THE  OPTICAL   PROPERTIES 

OF 

CRYSTALS 

WITH  A  GENERAL  INTRODUCTION  TO  THEIR 
PHYSICAL  PROPERTIES 

BEING  SELECTED  PARTS  OF  THE 

PHYSICAL    CRYSTALLOGRAPHY 

BY 

P.  GROTH 
•^ 

Professor  of  Mineralogy  and  Crystallography  in  the 
University  of  Munich 

TRANSLATED  (WITH  THE  AUTHOR'S  PERMISSION)  FROM  THE 

FOURTH,  REVISED  AND  AUGMENTED 

GERMAN  EDITION 

BY 
B.   H.   JACKSON,    M.E.,   M.A. 

University  of  Colorado 

TOUtb  121  Jfigures  in  tbc  Ueit  anl>  Uwo  Coloreb  plates 


FIRST  EDITION 

FIRST    THOUSAND 


NEW  YORK 

JOHN    WILEY    &    SONS 

LONDON:    CHAPMAN  &   HALL,  LIMITED 

1910 


COPYRIGHT,  1910, 

BY 
B.  H.  JACKSON 


Stanbopc  jpreas 

p.  H.  GIl-SON  COMPANY 
BOSTON.  U.S.A. 


PREFACE 


UNTIL  recently,  in  the  higher  institutions  of  learning,  crys- 
tallography has  largely  been  taught  only  in  connection  with 
mineralogy,  as  an  aid  to  the  characterization  of  minerals  and 
therefore  in  a  purely  descriptive  manner.  But  this  does  not 
accord  with  the  present  state  of  the  science.  Haiiy,  the 
founder  of  crystallography,  had  already  made  an  attempt  to 
explain  the  forms  of  crystals,  while  the  investigations  of  Brewster 
and  the  later  ones  of  Senarmont  and  Grailich,  together  with 
those  conducted  recently  by  Mallard  and  others,  have  given 
us  a  detailed  knowledge  of  the  regular  connection  between  the 
physical  properties  of  crystals  and  the  crystal  form.  As  a 
consequence  of  these  discoveries  the  conviction  has  gradually 
made  its  way  that  the  form  of  a  crystal  is  solely  a  consequence 
of  its  interior  structure,  —  of  its  make-up  from  the  smallest 
crystal  particles,  which  act  on  one  another  with  definite  forces 
depending  regularly  on  the  crystallographic  direction,  —  and  is 
therefore  a  physical  property  of  the  substance  in  question. 
Hessel,  and  later  Bravais  and  Gadolin,  independently,  suc- 
ceeded in  determining  the  entire  number  of  possible  crystal 
forms  by  purely  geometrical  methods;  while  reasoning  based 
on  the  physical  properties  of  crystals  leads  to  exactly  the 
same  results.  For  the  conclusions  as  to  the  interior  structure  of 
crystallized  media  —  as  set  forth  in  the  theories  of  Bravais, 
Sohncke,  Fedorow,  Schonfliess,  and  others  —  that  necessarily 
follow  on  this  basis,  point  to  the  existence  of  exactly  the  same 
kinds  of  symmetry.  The  consequence  is  that  crystallography 


240773 


VI  PREFACE 

must  be  regarded  as  a  part  of  molecular  physics;  and,  since 
Voigt  has  made  it  seem  probable  that  the  so-called  "amor- 
phous" bodies  are  to  be  conceived  of  as  aggregates  of  very 
small  crystalline  particles,  the  science  may  be  designated  quite 
generally  as  the  "molecular  physics  of  solids". 

A  scientific  treatment  of  this  subject,  then,  can  proceed  only 
hand  in  hand  with  the  entire  physics  of  crystals.  And  in  its 
theoretical  aspect  the  edifice  of  crystal  lore  stands  at  the  present 
day  as  one  of  the  best  established  in  the  whole  realm  of  physics, 
—  of  fundamental  importance  for  an  understanding  of  the 
material  world. 

But  not  only  in  theoretical  but  also  in  practical  respects  has 
the  treatment  of  crystal  science  by  physical  methods  attained  a 
constantly  increasing  importance.  Our  complete  knowledge  of 
the  regular  relation  between  the  optical  properties  and  the 
symmetry  of  crystals  has  given  us  the  means  to  carry  out,  by 
optical  methods,  the  determination  of  crystallized  substances  in 
microscopic  preparations  with  a  certainty  which,  only  a  few 
decades  ago,  no  one  would  have  believed  possible.  It  is  com- 
monly known  what  a  revolution  petrography  has  experienced  in 
consequence  of  this,  and  how  important  an  aid  to  the  chemist 
"crystal- analysis"  has  become  through  the  work  of  O.  Lehmann, 
C.  Haushofer,  and  others;  it  is  well  known,  too,  how  fruitful 
have  been  the  methods  of  crystal  optics  in  botanical  and  his- 
tologic-zoological  investigations. 

Under  these  circumstances  students  of  natural  science, 
especially  those  devoting  themselves  to  chemistry,  miner- 
alogy, and  geology,  can  no  longer  be  permitted  to  neglect  the 
study  of  crystallography  in  the  sense  indicated.  The  present 
text-book  presumes  an  acquaintance  with  general  experimental 
physics  and  chemistry,  but  with  no  mathematics  beyond  what 
is  afforded  by  secondary  schools;  it  aims  not  only  to  lead  the 
student  to  an  understanding  of  the  laws  to  which  crystallized 
substances  are  subject,  but  also  to  enable  him  to  turn  the 
methods  of  the  science  to  their  practical  application.  No 


PREFACE  Vll 


difficulties  of  any  sort  should  be  met  with,  especially  by  those 
who  have  attended  a  lecture  on  mineralogy  and  gained  thereby 
an  idea  of  the  most  common  crystal  forms. 


In  the  fourth  edition  of  "  Physikalische  Krystallographie", 
in  describing  the  properties  of  crystallized  bodies  in  general,  a 
system  has  been  introduced  which  makes  it  possible  to  place 
clearly  before  beginners  the  laws  governing  the  dependence  of 
the  crystal  properties  on  the  crystallographic  direction,  in  a 
steady  advance  from  simple  to  complex.  While  in  respect  of 
their  optical,  thermal,  electrical,  and  magnetic  behavior,  the 
totality  oi  crystals  fall  into  only  Jive  groups,  this  number  is 
increased  by  the  properties  of  cohesion  and  elasticity  to  seven 
and  nine  respectively;  and  the  behavior  of  crystals  with  regard 
to  solution  and  growth  gives  us,  finally,  all  the  possible  sym- 
metry classes  of  crystals,  —  numbering  thirty-two.  The  most 
important  part  of  the  subject  from  a  practical  standpoint  is 
that  concerned  with  optics.  Here,  in  consequence  of  Lord 
Kelvin's  researches  and  Fletcher's  lucid  exposition,  the  un- 
tenable "  elasticity"  of  the  ether  had  already  been  laid  aside 
in  the  third  edition.  The  present  treatment  of  crystal  optics 
is  based  on  its  purely  geometrical  aspect,  suggested  by  the 
latter  author.  This  method  makes  it  possible,  yet  without 
mathematical  theory,  to  gain  a  correct  insight  into  even  the 
most  complicated  phenomena  (conical  refraction,  for  example), 
something  which  is  indispensable  for  microscopical  studies. 
Other  changes  are  limited  essentially  to  particular  amplifica- 
tions —  an  example  of  which  occurs  in  the  discussion  of  total 
reflection  —  and  to  making  the  wording  more  precise.  —  Excerpts 
(with  slight  adaptations  by  the  translator)  from  the  prefaces  to  the 
third  (1895)  and  fourth  (1905)  German  editions. 


TRANSLATOR'S   NOTE 


THIS  partial  translation  of  the  fourth  edition  of  Groth's 
"  Physikalische  Krystallographie "  is  made  up  chiefly  of  matter 
contained  in  Part  I  of  the  original  work,  —  on  "The  Properties 
of  Crystals";  besides  embracing  the  general  introduction  and  all 
that  falls  under  the  heading  "  Optical  Properties  "  in  this 
part,  it  includes,  also  whatever  may  be  found  there  on  the  influ- 
ence of  other  properties  on  the  optical  properties.  Short  extracts 
from  Parts  II  ("Systematic  Description  of  Crystals")  and  III 
("The  Methods  of  Crystal  Investigation")  have  been  introduced, 
on  occasion,  for  illustration  and  example. 

The  translation  was  undertaken  at  the  instance  of  Professor 
Russell  D.  George,  of  the  University  of  Colorado;  the  translator 
is  indebted  to  him  for  his  kindly  interest  and  frequent  advice, 
as  well  as  to  Professor  Oliver  C.  Lester  for  several  important 
suggestions. 

While  the  author  has  elsewhere  been  closely  followed,  it  was 
found  expedient,  in  discussing  the  influence  of  other  properties 
on  the  optical  properties,  to  select  the  parts  to  be  translated  and 
sometimes  to  condense  them,  as  well  as  to  make  considerable 
changes  in  their  arrangement  and  classification.  The  confusing, 
tautological  expression  "  direction  of  "  an  optic  axis,  common  in 
both  languages,  has  been  avoided.  Additional  references  and 
cross-references  have  sometimes  been  given,  and  a  complete  table 
of  abbreviations  is  appended;  any  further  amplifications  by  the 
translator  will  be  found  enclosed  in  brackets  or  otherwise 
indicated. 

BOULDER,  COLORADO, 
March,  1910. 

ix 


ABBREVIATIONS 


Abh.  Ges.  d.  Wissensch.  Gottingen.  —  Abhandlungen  der  k.  Gesellschaft  der  Wissen- 
schaften  zu  Gottingen.    Berlin. 

Bull.  soc.  franc,  de  mineral.  —  Bulletin  de  la  Societe  francaise  de  mineralogie. 
Paris. 

Carls  Repert.  f.  Exper.-Physik.   (Carls)   Repertorium   fiir  Experimental-Physik. 
Leipzig. 

Centralbl.  f.  Min.  —  Centralblatt  fur  Mineralogie.     Stuttgart. 

Leiss:  Die  op.   lustrum.  —  C.    Leiss:    "  Die   optischen   Instrumente   der   Firma 
Fuess."     Leipzig,  1899. 

Min.  Mag.  —  Mineralogical  Magazine  (and  Journal  of  the  Mineralogical  Society). 
London. 

Nachr.  Ges.  d.  Wissensch.  Gottingen.  —  Nachrichten  der  k.  Gesellschaft  der  Wissen- 
schaften  zu  Gottingen.     Gottingen. 

Phil.  Mag.  —  London,  Edinburgh,  and  Dublin  Philosophical  Magazine.     London. 

Phys.    Kryst.  —  P.  Groth:  "  Physikalische  Krystallographie."     Leipzig;  3rd  ed. 
1895,  4th  ed.  1905. 

Pogg.  Ann.  d.  Physik.  —  (Poggendorff's)  Annalen  der  Physik.     Leipzig. 
Proceed.  Phys.  Soc.  —  Proceedings  of  the  Physical  Society.     London. 

Sitzungsber.  Akad.  d.  Wissensch.  Berlin.  —  Sitzungsberichte  der  k.  p.  Akademie  der 
Wissenschaften  zu  Berlin.     Berlin. 

Tscher.  min.   u.   petrog.   Mitteil.  —  (Tschermak's)    mineralogische  und   petrogra- 
phische  Mitteilungen.     Vienna. 

Zeitschr.  f.  Kryst.  —  Zeitschrift  fur  Krystallographie.     Leipzig. 

Zeitschr.  f.    Kryst.    u.   Min.  —  (Groth's)   Zeitschrift  fur  Krystallographie  und 
Mineralogie.     Leipzig. 


XI 


TABLE    OF    CONTENTS 


PAGE 

General  Introduction  to  the  Properties  of  Crystals 3 

THE    OPTICAL    PROPERTIES   OF   CRYSTALS 

The  Nature  of  Light n 

Combination  (Interference)  of  Plane-polarized  Light 16 

Optically  Isotropic  (Singly  Refracting)  Bodies 

Propagation  of  Light 21 

Reflection  of  Light 26 

Refraction  of  Light 30 

Polarization  of  Light  by  Reflection  and  Refraction 49 

Double  Refraction  of  Light 50 

Polarization-colors  of  Doubly  Refracting  Crystals 63 

Polarization  Apparatus 74 

Optically  Uniaxial  Crystals 

Double  Refraction  of  Light  in  Calcite 81 

Double  Refraction  of  Light  in  Other  Uniaxial  Crystals 101 

Behavior  of  Uniaxial  Crystals  in  the  Polarization  Apparatus 106 

Optically  Biaxial  Crystals 

Deduction  of    the  Optical    Properties    of    Crystals   from    a  Surface  of 

•Reference  (Optical  Index-surface  or  Indicatrix) 121 

Ray-surface  of  the  Optically  Biaxial  Crystals 129 

Determination  of  the  Principal  Refractive  Indices  of  Biaxial  Crystals 144 

Interference  Phenomena  of  Biaxial  Crystals  in  Parallel  Polarized  Light  152 

Interference  Phenomena  of  Biaxial  Crystals  in  Convergent  Polarized  Light  161 
Determination  of  the  Optic  Axes  in  Biaxial  Crystals  and  Measurement 

of  their  Angle 183 

Recapitulation:  Classification  of  Crystals  According  to  their  Optical  Properties  196 

Combinations  of  Doubly  Refracting  Crystals 

Determination  of  the  Character  of  the  Double  Refraction  of  Uniaxial 
and  Biaxial  Crystals  by  Combination  with  Other  Doubly  Refract- 
ing Crystals 198 

Optical  Behavior  of  Combinations  of  Doubly  Refracting  Crystals  of  the 

Same  Kind 211 

Rotation  of  the  Polarization  Plane  of  Light  in  Crystals 220 

xiii 


XIV  TABLE  OF  CONTENTS 

PAGE 

Absorption  of  Light  in  Crystals 232 

Brush  Phenomena '. 246 

Surface  Colors 249 

Fluorescence 251 

Phosphorescence 252 

Influence  of  Other  Properties  on  the  Optical  Properties  of  Crystals 

Thermal  Properties 253 

Elastic  Strain  by  Mechanical  Forces 

Homogeneous  Strain 261 

Elastic  Strain  Not  Homogeneous 262 

Optically  Anomalous  Crystals 279 

Elastic  Strain  by  Electrical  Action 282 

Permanent  Strain 

Plasticity 283 

Gliding 284 

Twinning  . 288 

APPENDIX 

Supply  Houses  for  Apparatus,  Models,  Crystals,  and  Preparations 293 

INDEX 299 


ERRATA 

Page  5,  sixth  line  from  bottom  of  page,  for  "  composition  "  read 
constitution."  * 

Page  264,  next  to  last  line  of  footnote,  for  "  negatively  "  read 
positively."'' 


XIV 


TABLE  OF   CONTENTS 


Absorption  of  Light  in  Crystals 232 

Brush  Phenomena '. 246 

Surface  Colors 249 

Fluorescence 251 

Phosphorescence 252 

Influence  of  Other  Properties  on  the  Optical  Properties  of  Crystals 

Thermal  Properties 253 

Elastic  Strain  by  Mechanical  Forces 

Homogeneous  Strain 261 

Elastic  Strain  Not  Homogeneous 

Optically  Anomalous  Crystals 

Elastic  Strain  H»  ^' 


GENERAL  INTRODUCTION  TO  THE  PROPERTIES 
OF  CRYSTALS 


GENERAL   INTRODUCTION 
TO  THE  PROPERTIES  OF  CRYSTALS 

WHEN  a  body  has  the  same  constitution  at  all  points,  so  that 
any  two  equal,  similar,  and  similarly  oriented  parts  of  it  are 
undistinguishable  from  each  other  by  any  difference  in  quality, 
that  body  is  said  to  be  homogeneous. 

Homogeneous  bodies  fall  into  two  classes:  — 

1.  Bodies  in  which  not  only  all  points,  but  also  all  directions, 
are  equivalent;  i.e.  in  which  the  different  directions  are  undis- 
tinguishable from  one  another  by  any  physical  property  of  the 
body.     These  bodies  are  spoken  of  as  amorphous,  because  they 
have  no  peculiar  shape,  or  as  isotropic,  because  they  transmit 
every  kind  of  motion  in  the  same  way  in  all  directions.     Here 
belong  all  gases  and  vapors,  and  nearly  all  liquids;   also  a  num- 
ber of  so-called  "solids",  as  colloids,  resins,  glasses.     But  the 
latter  bodies  are  not  sharply  separated  from  the  liquids;    for 
example,  with  an  increase  of  temperature  they  pass  through  the 
softened,  or  viscous,  state  gradually  over  into  the  liquid. 

2.  Homogeneous   bodies  whose   properties   depend   on   the 
direction,  so  that  the  value  of  any  one  property  attains  in  certain 
directions  a  maximum,  in  others  a  minimum.     (This  may  be 
the  case  only  for  particular  physical  properties,  while  for  others 
there  may  exist,  grounded  in  the  nature  of  the  property  in  ques- 
tion, an  equality  of  the  value  for  all  directions.)     Bodies  of  this 
kind  are  capable  of  crystallization,  — i.e.  of  assuming  a  regular 
form  which  is  peculiar  to  the  body  and  which  stands  in  a  regular 
relation  to    the   non-equivalence  of    the  directions  within  it  — 
and  are  therefore  called  crystallized  or  crystalline  bodies. 

3 


4  GENERAL   INTRODUCTION 

The  physical  properties  are  distinguished  into  scalar  and 
vector. 

The  scalar  properties  are  properties  represented  by  a  single 
quantity  independent  of  the  direction,  as  temperature,  density, 
specific  heat,  etc. 

The  vector  properties  are  such  as  are  denned  by  a  numerical 
value  and  a  direction.  When  the  numerical  value  is  necessarily 
the  same  in  the  opposite  direction,  i.e.  when  the  two  directions 
that  pass  out  from  any  point  and  belong  to  the  same  straight 
line  are  always  absolutely  equivalent  in  respect  of  a  property, 
then  that  property  is  designated  as  bi-vector.* 

From  the  foregoing  definitions  it  follows  that  the  amorphous 
bodies  can  possess  only  scalar  properties,  while  the  crystallized, 
on  the  other  hand,  have,  besides  these,  vector  and  bi-vector 
properties. 

To  set  forth  the  regular  relations  that  exist  among  the  crystal 
properties  depending  on  the  direction  — and  with  those  proper- 
ties belongs  the  geometric'  form  —  is  the  object  of  physical 
crystallography.  A  plausible  explanation  of  the  laws  of  this 
branch  of  science  is  supplied  us  by  the  molecular  hypothesis,  if 
we  assume  that  within  crystals,  while  the  molecules  are  indeed 
in  motion,  yet  their  motion  consists  of  vibrations  about  certain 
intermediate  loci,  and  that  these  loci  are  regularly  arranged  in 
space.  |  Then  the  manner  of  this  arrangement,  which  is  known 

*  Instead  of  the  usual  designation  "bi-vector",  Voigt  has  proposed  the  use 
of  "tensor";  but  for  the  purposes  of  crystallography  the  former  term  seems  the 
more  representative.  , 

t  In  contradistinction  from  this,  for  the  amorphous  bodies  there  must  be 
assumed  an  irregular  distribution  of  the  molecules  in  space,  something  which  for 
gases  and  liquids  is  at  once  clear.  An  amorphous  body,  according  to  this  second 
assumption,  is  only  apparently  homogeneous.  That  is,  its  unhomogeneousnesses, 
because  of  their  too  rapid  succession  within  the  smallest  compass,  are  no  longer 
accessible  to  physical  examination;  and  the  latter  therefore  yields  for  every 
property  only  a  mean  value,  which,  naturally,  is  found  to  be  the  same  for  all 
directions.  For  this  reason  crystals  have  even  been  designated  as  the  only  really 
homogeneous  bodies,  and  the  terms  "crystallized"  and  "homogeneous"  as  equiv- 
alent. 


GENERAL  INTRODUCTION  5 

as  crystal  structure,  corresponds  to  a  state  of  stable  equilibrium 
among  the  interior  forces.  But,  since  this  equilibrium  is  influ- 
enced by  the  vibrations  depending  on  the  heat  content  of  the 
body,  the  arrangement  corresponding  to  the  more  stable  equilib- 
rium may  for  other  temperatures  be  different.  In  this  case  the 
crystallization  under  other  conditions  of  temperature  and  pres- 
sure results  in  an  arrangement  other  than  the  first ;  and  if  a  certain 
critical  temperature  is  transgressed  there  will  come  to  pass  a 
transformation  of  the  first  crystallized  body  into  a  second,  chem- 
ically the  same  as  the  first  but  physically  different,  just  as  at  the 
so-called  melting,  or  freezing,  point  a  transformation  takes  place 
from  the  crystalline  state  into  the  amorphous,  or  vice  versa. 
Precisely  as  the  melting,  or  freezing,  point  can  be  overstepped 
without  transformation,  — the  state  of  the  body  then  becoming 
labile,  —  so  too  does  the  same  occur  in  the  transformation  of  the 
different  crystalline  states,  or  "modifications",  into  one  another. 
The  property  of  a  substance  to  present  itself  in  several  modifica- 
tions is  termed  polymorphism  (dimorphism,  trimorphism,  etc.), 
or,  to  distinguish  it  from  the  chemical  isomerism,  physical 
isomerism.  The  differences  among  the  polymorphous  modifi- 
cations of  a  body  exist  only  in  the  crystalline  state:  the  transition 
into  the  amorphous  state  (by  fusion  or  vaporization,  and  also  by 
solution)  necessarily  does  away  with  them. 

Since,  according  to  the  foregoing  assumptions,  the  crystal 
structure  of  a  body  depends  on  the  nature  of  its  molecules,  there 
must  exist  a  regular  relation  between  the  crystal  structure  and 


the  chemical  eem^SBfE&i  of  the  body;  to  set  forth  this  relation  is 
the  object  of  chemical  crystallography.*  This  study  teaches 
that  two  chemically  related  bodies  can  present  themselves  in 
crystalline  modifications  whose  respective  structures  stand  in 
close  relationship  to  one  another,  so  that  it  is  possible  to  deter- 
mine the  variation  in  crystal  structure  that  is  produced  by 

*  A  condensed  statement  of  the  subject  is  given  by  the  author  in  his  "Intro- 
duction to  Chemical  Crystallography  ".  Authorized  translation  by  Hugh  Mar- 
shall. Edinburgh  and  New  York,  1906. 


6  GENERAL   INTRODUCTION 

a  change  in  the  composition  of  the  molecule;  such  relation- 
ships are  designated  as  morpho tropic.  They  are  most  intimate 
(always  supposing  that  corresponding  modifications  of  the 
different  substances  are  taken  for  the  comparison)  when  it  is  a 
matter  of  two  bodies  whose  respective  molecules  differ  only  in 
this:  that  into  the  place  of  an  atom  in  the  molecule  of  the  first 
body  there  has  entered,  in  the  case  of  the  second,  an  atom  of 
different  element,  which  has  the  same  valence  as  the  first  ele- 
ment and  is  very  closely  related  to  it.  In  such  cases  the  crystal 
structure  of  the  two  bodies  is  so  similar  that  their  crystal  form 
is  very  nearly  the  same,  under  some  circumstances  even  abso- 
lutely identical;  such  crystallized  substances  are  therefore  said 
to  be  isomorphic.  Isomorphic  bodies  are  capable  of  crystalliz- 
ing together;  i.e.  of  mixing  together  in  various  proportions  to 
form  crystals  (so-called  isomorphic  mixtures)  which  behave, 
physically,  like  homogeneous  bodies  and  whose  properties  vary 
continuously  with  the  composition. 

In  order  to  present  the  regular  connection  among  the  crystal 
properties  depending  on  the  direction,  these  properties  must  be 
brought  into  a  system  from  which  the  individual  regularities 
follow  as  a  matter  of  necessity.  The  basis  of  this  system  is 
symmetry.  When  a  body  is  so  constituted  that  the  two  opposite 
directions  passing  out  from  any  point  are  absolutely  equivalent, 
we  say  it  has  a  "center  of  symmetry";  therefore  the  bi-vector 
properties  are  designated  also  as  "centrally  symmetrical",  and, 
in  contradistinction  from  them,  the  vector  as  "acentric". 

The  bi-vector  properties  are  further  distinguished  into  those 
of  higher  symmetry  and  those  of  lower  symmetry. 

The  former  have  in  common  that  for  any  and  every  direction 
their  numerical  value  is  determined  by  three,  at  the  most, 
numerical  quantities,  which  refer  respectively  to  three  definite, 
mutually  perpendicular  directions;  and  the  numerical  value  of 
a  definite  property  of  this  kind,  for  any  direction,  is  proportional 
to  the  radius  vector  corresponding  to  that  direction,  of  a  triaxial 
ellipsoid  whose  three  semi-axes  are  proportional  to  the  three 


GENERAL   INTRODUCTION  7 

numerical  quantities  holding  good  with  the  property.  The 
"bi-vector  properties  of  higher  symmetry"  are  therefore  called 
also  ellipsoidal  properties.  In  the  special  case  where  the  throe 
principal  axes  of  the  ellipsoid  are  of  equal  length,  the  ellipsoid 
passes  over  into  a  sphere;  that  is,  the  numerical  value  of  the 
property  in  question  is  independent  of  the  direction,  like  that  of 
a  scalar  property.  For  this  reason  it  is  with  the  ellipsoidal 
properties  that  the  relations  are  simplest,  wherefore  the  pres- 
entation of  the  subject  begins  most  properly  with  them;  and,  of 
them,  with  the  optical  properties.  These,  on  account  of  their 
practical  importance,  will  not  only  be  developed  from  the  funda- 
mental ideas,  but  also  treated  so  far  in  detail  as  is  demanded  by 
the  practical  application  of  optical  methods  to  the  determination 
of  crystalline  forms  in  chemical  crystallography,  mineralogy, 
petrography,  and  other  natural  sciences.  The  remaining  ellip- 
soidal properties,  i.e.  the  thermal,  the  electrical,  and  the  magnetic, 
are  so  absolute  analogous  to  the  optical  that  they  need  be 
presented  only  in  brief.* 

With  the  properties  of  elasticity  and  cohesion  of  crystals  the 
relations  are  more  complex:  the  dependence  of  these  prop- 
erties on  the  direction  cannot  be  represented  by  a  surface  of 
such  simple  form  as  an  ellipsoid  or  a  sphere,  but  only  by  one 
of  more  complicated  form  and  of  an  in  general  lesser  degree 
of  symmetry.  These  properties  shall  therefore  be  distinguished 
from  those  of  higher  symmetry,  the  ellipsoidal  properties,  as 
properties  of  lower  symmetry. 

The  lowest  degree  of  symmetry,  finally,  is  possessed  by  the 
vector  properties,  the  properties  in  respect  of  which  even  the 
two  opposite  directions  pertaining  to  the  same  straight  line  are 
not  necessarily  equivalent.  Among  these  crystal  properties 
belong  those  of  solution  and  growth;  the  latter  are  the  propert'es 
in  virtue  whereof  the  crystal  assumes  its  definite  geometric  form. 

Those  bi-vector  and  those  vector  properties  with  which  it  is 

*  [This  refers  of  course  to  the  original  work,  as  the  present  translation  deals 
only  with  the  optical  properties.] 


8  GENERAL  INTRODUCTION 

a  matter  of  the  action  of  mechanical  forces  on  the  crystal,  are  of 
especially  high  theoretical  significance  for  the  reason  that,  with 
the  crystal  under  such  conditions,  the  forces  must  be  overcome 
that  are  exerted  by  its  smallest  particles  on  one  another.  The 
consideration  of  these  properties  therefore  leads  to  that  of 
the  causes  of  crystal  structure,  and  thus  to  the  consideration  of 
these  particles  and  forces  themselves.  For  treating  the  theories 
of  crystal  structure  that  come  into  consideration  there  are 
requisite  certain  conceptions  from  geometry*;  and  these  con- 
ceptions are  of  additional  importance  for  the  reason  that,  being 
applicable  likewise  to  the  theories  respecting  the  configuration 
of  the  atoms  in  the  molecules,  they  may  be  considered  as  the 
basis  of  stereochemical  views.  These  conceptions  shall  there- 
fore first  of  all  be  elucidated,  and  then,  with  their  aid,  all  the 
laws  deduced  that  govern  the  geometric  form  of  crystals;  while 
the  particulars  of  the  several  classes  of  symmetry,  which 
follow  from  all  those  general  laws,  and  the  description  of  a 
number  of  crystallized  bodies  forming  especially  important  or 
interesting  examples  of  the  several  classes,  are  reserved  for 
Part  II.| 

*  According  to  this,  geometry  is  an  auxiliary  science  to  crystallography,  not 
the  latter  a  part  of  the  former;  for  geometry  deals  with  the  form,  crystallography 
with  the  contents  of  the  same,  —  i.e.  with  the  material  of  the  crystal  as  carrier  of 
the  properties,  among  which  the  form  belongs. 

t  **  Systematische  Beschreibung  der  Krystalle."    (Cf.  footnote  on  p.  7.) 


THE  OPTICAL  PROPERTIES  OF  CRYSTALS 


THE   OPTICAL   PROPERTIES   OF 
CRYSTALS 

THE    NATURE    OF    LIGHT 

To  explain  the  properties  of  light  we  assume  it  is  a  periodic 
motion  of  the  smallest  particles  of  the  luminiferous  ether,  a  form 
of  matter  which  pervades  universal  space  and  likewise  all  bodies, 
but  which  within  the  latter,  under  the  influence  of  ponderable 
matter,  takes  on  certain  peculiarities. 

If  we  imagine  the  ether  at  rest,  i.e.  the  forces  acting  among 
its  particles  in  equilibrium,  and  if  by  an  impulse  a  particle  be 
removed  from  the  position  corresponding  to  this  equilibrium,  if 
it  thus  be  given  a  certain  velocity  in  any  direction,  then  accord- 
ing to  this  theory  a  force  is  awakened;  this  force  is  such  that  it 
drives  the  particle  back  toward  its  original  position,  thus  dimin- 
ishing its  velocity,  and  that  in  exact  accordance  with  the  same 
law  after  which  gravity  acts  on  a  pendulum  set  in  motion  by  a 
sudden  push.*  The  ether  particle  will  therefore  at  a  certain 
distance  from  its  original  position  acquire  the  velocity  zero,  then 

*  After  Huygens,  the  originator  of  the  theory  of  the  luminiferous  ether,  this 
force  is  commonly  designated  as  elastic,  and  therefore  we  speak  of  the  "elasticity 
of  the  ether";  yet  it  must  be  remarked  that  to  explain  the  phenomena  of  light 
no  further  assumption  is  requisite  than  the  quite  general  one  mentioned  above  — 
that  by  a  displacement  of  the  ether  particle  a  force  is  awakened  which  produces  a 
motion  corresponding  to  the  laws  of  the  pendulum.  For  this  property  of  the 
ether,  on  which  the  transmission  of  light  depends,  Fletcher  ("The  Optical  Indica- 
trix  and  the  Transmission  of  Light  in  Crystals."  London,  1892,  p.  97)  proposes 
the  name  "resilience".  If  one  imagines  in  parallel  position  a  row  of  pendulums 
among  which  there  exists  a  connection  representing  that  force,  and  imparts  a 
motion  to  one  of  them,  one  obtains  the  complete  analogue  of  the  motion  of  a  row 
of  ether  particles,  described  in  the  following. 


12  THE  NATURE  OF  LIGHT 

return,  under  the  influence  of  that  force,  with  increased  velocity, 
and  reach  the  position  of  equilibrium  with  the  same  velocity  it 
received  from  the  initial  impulse,  but  in  the  opposite  direction; 
so  that  from  the  original  position  it  now  moves  in  the  same  way*, 
but  in  the  other  direction,  with  diminishing  velocity,  in  conse- 
quence of  the  force  mentioned,  to  the  same  distance  as  before, 
and  thence  returns  with  increasing  velocity.  When  finally  the 
ether  particle  has  again  arrived  at  the  point  where  its  motion 
began,  it  is  moving  in  the  same  direction  as  in  the  beginning, 
with  exactly  the  same  velocity;  it  therefore  begins  a  second 
vibration,  as  such  a  to-and-fro  motion  is  called.  The  extent  of 
the  vibration,  the  distance  the  particle  travels  during  the  same, 
is  called  its  maximum  amplitude,  or  briefly  its  amplitude  * ;  the 
time  required  for  the  execution  of  one  whole  vibration,  i.e.  up  to 
the  next  return  of  the  same  state  of  vibration,  is  called  the  time 
of  vibration,  or  the  period.  The  amplitude  of  the  vibration 
depends  on  the  velocity  with  which  the  particle  is  moved  from 
the  position  of  rest,  and  is  proportional  to  that  velocity;  for 
example,  if  the  initial  impulse  is  such  that  it  imparts  double 
the  initial  velocity,  then  in  the  same  time  the  particle  travels 
double  the  distance  and  so  attains  double  the  maximum  ampli- 
tude, while  the  period  remains  constant. 

If  the  ether  is  a  homogeneous  medium  in  the  most  proper 
sense  (cf.  footnote  f>  P-  4)>  then  on  a  straight  line  its  particles 
must  stand  at  equal  distance  from  one  another,  and  this  dis- 
tance must  correspond  to  equilibrium  of  the  forces  (of  attraction 

and    repulsion)    which    act    between 
*     /3      y      d      e       i 

•  ->  •      •      •      •      each    particle   and    the    surrounding 

i  *  particles.     Hence,  if  we  imagine  one 

Fi    i  of  the  particles  of  such  an  equidis- 

tant row,  e.g.  a,  Fig.  i,   as  set  into 

the  described  vibratory  motion,  for  example,  if  a  be  given  a 
velocity  in  the  direction  of  a',  its  distance  from  /?  is  increased, 

*  [Amplitude  is  more  commonly  defined  as  the  extent  of  the  excursion  of  the 
particle  on  either  side  of  its  position  of  equilibrium.] 


THE  NATURE   OF  LIGHT  13 

and  in  consequence  of  the  force  thereby  awakened  the  particles 
a  and  /?  must  try  to  approach  each  other:  not  only  is  a,  in  its 
new  position  a',  drawn  back  by  /?,  but  also  /?  is  attracted  toward 
a/,  and  attracted  the  more,  the  greater  that  force  awakened  in 
the  ether.  But  in  virtue  of  the  same  force  the  motion  of  /? 
toward  a'  is  resisted  by  the  attraction  of  7-;  so  that  /?,  in  con- 
sequence of  these  two  attractions  of  a!  and  f  respectively,  takes 
an  intermediate  direction,  — namely,  that  parallel  to  aa' '.*  In 
the  same  way  the  particle  7-  is  hereupon  caused  by  the  motion  of 
y?  to  move  in  the  same  direction;  and  so  on,  all  the  following 
particles.  When  the  motion  has  been  transmitted  as  far  as  a 
certain  particle,  v  (Fig.  2),  just  beginning  its  motion,  the  row 


•  ••  :  5  ! 

Fig.  2. 

of  points,  previously  straight,  at  this  instant  forms  a  wave,  con- 
sisting of  a  crest  and  a  trough  of  equal  length.  Such  a  motion 
is  called  a  wave-motion;  the  distance  av,  the  distance  the  wave- 
motion  was  transmitted  while  the  first  particle  carried  out  one 
vibration,  is  called  the  wave  length  and  is  denoted  by  L  Within 
one  wave  length  are  present,  side  by  side,  all  states  of  vibration 
that  are  possessed  in  succession,  by  one  and  the  same  particle, 
during  the  time  of  one  vibration.  Every  two  particles  whose 
distance  apart  is  %X  are  in  the  opposite  state  of  vibration. 
The  quantity  X  is  obviously  proportional  both  to  the  vibration 
period  and  to  the  velocity  with  which  the  motion  is  transmitted. 
For  if  a  wave-motion  had  the  same  transmission  velocity,  but 
double  the  vibration  period,  it  would  advance  double  the  dis- 
tance while  the  first  particle  executed  one  vibration;  but  like- 
wise would  X  become  twice  as  great  if,  with  the  same  vibration 

*  Since  the  displacement  aa'  is  infinitesimally  small,  the  angle  between  a'ft  and 
/?/•  is  to  be  regarded  as  infinitesimally  less  than  180°. 


14  THE   NATURE   OF   LIGHT 

period,  the  transmission  of  the  motion  took  place  with  double 
the  speed. 

Supposing  the  motion  to  have  been  transmitted  through  a 
longer  row  of  points,  these  now  form  a  wave  system,  which 
is  divided  into  a  number  of  wave  lengths,  all  equal  if  the 
relations  on  the  whole  row  of  points,  and  accordingly  also  the 
transmission  velocity  of  the  motion,  remain  the  same.  All  the 
particles  of  the  wave  system  that  stand  apart  from  one  another 
by  ^  or  by  a  whole  multiple  of  ^  are  in  the  same  state  of  vibra- 
tion; on  the  other  hand,  all  those  whose  distance  apart  amounts 
to  \X  or  to  any  uneven  multiple  of  %X  are  in  the  opposite  state 
of  vibration.  But  of  course  such  a  wave  system  represents  the 
state  of  a  row  of  ether  particles  only  at  some  definite  instant: 
at  every  subsequent  instant  all  the  particles  are  in  one  of  the 
succeeding  states  of  vibration,  according  to  the  velocity  with 
which  the  wave-motion  advances. 

If  one  imagines  the  row  of  ether  particles  considered  on  pages 
12  and  13  as  continued  in  the  same  way  on  the  left  of  a,  then 
the  same  consideration  must  apply  to  all  the  particles  here  follow- 
ing a;  that  is  to  say,  if  at  any  point  of  such  a  row  a  light-motion 
is  excited,  it  must  be  transmitted  out  from  that  point  in  the  two 
opposite  directions  in  the  same  way.  On  this  depends  the  mem- 
bership of  the  optical  properties  among  the  bi-vector  and  the 
circumstance  that  every  optical  construction  applies  in  the 
reverse  sense. 

A  light  ray,  according  to  the  above  theory,  the  so-called 
"Undulatory  Theory  of  Light",  is  thus  a  wave  system;  i.e.  a 
straight  row  of  ether  particles  along  which  a  wave-motion  is 
transmitted  with  a  velocity  that,  although  finite,  is  very  great. 
The  amplitude  of  the  vibration  determines  the  brightness  of  the 
ray,  while  its  color  is  conditioned  by  the  period.  The  trans- 
mission velocity  of  light  in  empty  space  is  for  all  colors  the  same 
(about  186,000  miles  per  second) ;  and,  since  the  wave  length  X 
is  the  distance  the  wave-motion  is  transmitted  during  the  time 
of  one  vibration,  the  wave  length  of  light  for  different  colors 


THE   NATURE   OF   LIGHT  15 

must  be  exactly  proportional  to  the  vibration  period.  This 
latter  is  greatest  for  red  rays;  less  for  yellow,  green,  blue; 
and  least  for  violet  light,  which,  accordingly,  carries  out  its 
vibrations  the  most  rapidly.  The  wave  length  of  red  light  in 
empty  space  amounts  to  about  0.000760  mm.  and  that  of  violet 
to  about  0.000400  mm.;  it  follows  that  a  red  ray  must  execute 
about  390  bill  on  single  vibrations  in  one  second,  a  violet  on  the 
other  hand  about  750  billion. 

As  mentioned,  the  ether  pervading  ponderable  bodies  takes 
on,  under  their  influence,  other  properties,  in  virtue  whereof  it 
transmits  a  light-motion  with  another  velocity  (usually  less  than 
in  empty  space).  Consequently  THE  WAVE  LENGTH  OF  LIGHT 

OF   THE   SAME   COLOR   IS    DIFFERENT    IN    DIFFERENT    BODIES;    only 

the  vibration  period  remains  the  same;  this  it  is,  therefore,  that 
determines  the  specific  character  of  the  color. 

Besides  those  of  color  and  of  brightness,  we  observe  among 
light  rays  still  other  differences,  which  depend  on  the  manner  of 
the  vibratory  motion.  If  this  takes  place  as  in  the  example 
assumed  on  page  12,  along  a  definite  direction  perpendicular  to  the 
ray,  in  which  case  it  is  said  to  be  "transverse",  the  ray  in  ques- 
tion cannot  behave  alike  in  all  directions  about  the  line  along 
which  it  is  transmitted :  it  must,  since  all  the  vibrations  take  place 
in  one  definite  plane,  exhibit  a  certain  one-sidedness  (polarity) ;  it 
is  therefore  designated  as  a  plane-polarized  light  ray.  This  one- 
sidedness  must  obviously  be  symmetrical  with  reference  both  to 
the  plane  in  which  the  transverse  vibrations  take  place  and  to 
the  plane  that  intersects  this  plane  perpendicularly  in  the  trans- 
mission line  of  the  ray;  we.  shall  call  the  former  plane  the  trans- 
verse plane  of  the  ray,  the  latter  its  plane  of  polarization  (briefly : 
"polarization  plane").  Such  a  ray  can  only  then  behave  alike 
in  all  directions,  when  the  azimuth  of  its  transverse,  or  "vibra- 
tion", plane  rotates  very  rapidly  about  the  line  of  transmission; 
that  is  to  say,  when  the  successive  vibrations  always  take  place 
in  a  plane  that,  as  compared  with  the  plane  of  the  immediately 
preceding  vibration,  is  rotated  to  the  amount  of  a  small  angle,  so 


1 6  INTERFERENCE    OF    PLANE-POLARIZED    LIGHT 

that  after  a  certain  time  the  total  rotation  amounts  to  180°: 
during  this  time,  then,  the  vibrations  have  taken  place  in  all 
directions  about  the  ray.  Now  with  the  enormous  number  of 
light-vibrations  during  one  second  (see  above),  this  time  is  far 
shorter  than  is  required  to  apprehend  an  impression  of  light. 
During  the  latter  interval,  therefore,  the  vibration  plane  will 
have  rotated  many  times  about  the  line  of  transmission,  and 
the  light  ray  must  accordingly  produce  the  impression  of  iden- 
tical behavior  in  all  transverse  directions.  So-called  common, 
or  ordinary  (not  polarized),  light  consists  of  rays  whose  vibra- 
tory motions  are  of  the  kind  just  described. 

Besides  the  rectilinear  vibrations  there  are,  however,  light 
rays  of  other  kinds,  in  which  the  ether  particles  move  in  circles 
or  ellipses  and  which  consist  accordingly  of  circular  or  elliptical 
vibrations. 


COMBINATION  (INTERFERENCE)   OF  PLANE- 
POLARIZED    LIGHT 

If  an  ether  particle  is  caught  up  at  once  by  two  light-motions 
having  the  same  vibration  period,  it  arrives  after  a  certain  time 
at  the  point  to  which  the  single  motions  would  have  carried  it, 
had  they  acted  just  as  long  in  succession;  the  two  motions 
thus  combine  to  form  one  resultant.  The  simplest  case  of  this 
phenomenon  is  that  where  the  combining  wave-motions  are  trans- 
mitted along  the  same  direction  and  where  their  transverse  vibra- 
tions take  place  in  the  same  plane,  the  two  motions  differing 
therefore  only  in  state  of  vibration  and  in  amplitude;  in  this  case 
the  two  partial  motions  are  said  to  interfere,  and  their  combin- 
ing is  called  interference. 

If  from  a  source  of  light  there  proceed  simultaneously  light 
rays  having  parallel  vibration  direction  and  the  same  state  of 
vibration,  and  if  two  of  them  come  together  in  the  same  line 
of  transmission  after  they  have  passed  over  a  different  length  of 
path,  then  the  state  of  vibration  with  which  they  come  to  inter- 


INTERFERENCE    OF  PLANE-POLARIZED   LIGHT  17 

fere  at  a  point  of  their  now  common  path  is  in  general  unlike. 
The  difference  between  their  respective  states  of  vibration  fol- 
lows from  the  difference  between  the  lengths  of  path  they  have 
travelled,  expressed  in  wave  lengths.  This  difference  is  called 
their  difference  of  path*  (briefly:  "path  difference").  If  it 
equals  nX  (where  n  is  an  integer),  then  at  every  point  of  their 
now  common  path  the  two  wave  systems  interfere  with  the 
same  state  of  vibration.  Supposing  /  (Fig.  3)  to  represent  the 
state  in  which  a  row  of  ether  particles  would  be  at  a  certain 
instant  if  only  the  first  of  the  interfering  wave-motions  existed, 


X'" 


Fig.  3- 

and  //  their  state  at  that  instant  if  only  the  second  acted,  then  is 
///  the  state  of  the  ether  particles  in  consequence  of  the  wave- 
motion  resulting  from  the  two.  For  any  point,  x,  would  in 
virtue  of  the  first  partial  wave-motion  have  been  moved  to  #', 
in  virtue  of  the  second  to  x"\  so  it  must  now  be  at  a  distance 
xx"f  from  the  position  of  rest,  this  distance  being  the  sum 
of  the  displacements  xxf  and  xoc" ',  produced  by  the  single 
motions.  Accordingly,  there  results  from  the  interference  a 
wave-motion  which  has  the  same  state  of  vibration  as  the  com- 
bining motions  and  whose  amplitude  is  the  sum  of  their  ampli- 
tudes. 

When  two  interfering  light  rays  of  the  given  kind  come 
together  in  such  a  way  that  their  difference  of  path  amounts  to 

*  [Commonly  designated  by  crystallographers  as  "difference  in  phase".  But 
on  account  of  the  frequent  misapplication  of  the  word  'phase'  the  author,  in  his 
latest  edition,  has  discarded  it;  and  accordingly  has  made  no  service  of  the  term 
"Phasendifferenz",  using  instead  the  obviously  more  representative  "Gangunter- 
schied".] 


i8 


INTERFERENCE    OF   PLANE-POLARIZED    LIGHT 


i^  or  to  an  uneven  multiple  of  this  quantity,  then,  on  every 
ether  particle  caught  up  by  them  simultaneously,  they  act  in 
the  opposite  sense.  In  Fig.  4,  for  example,  the  point  y  would 
by  the  motion  /  alone  have  been  driven  upward  as  far  as  y', 
but  downward  to  y"  if  only  //  had  acted;  and  consequently, 
after  the  action  of  the  partial  motions  its  distance,  //",  from 
the  position  of  rest  must  be  the  difference  between  the  two  dis- 
tances. Thus,  from  the  interference  of  two  wave-motions  hav- 
ing \X,  |^,  or  f  \  (expressed  generally:  [n  +  J]  K)  difference 
of  path  there  results  a  single  wave-motion  (///  in  Fig.  4)  whose 


ii 


Fig.  4. 

state  of  vibration  is  the  same  as  that  of  the  stronger  of  the  two 
combining  wave-motions,  and  whose  amplitude  is  equal  to  the 
difference  between  their  amplitudes.  In  the  special  case  where 
the  two  interfering  light  rays  have  equal  amplitude,  i.e.  equal 
brightness,  that  of  the  resultant  is  zero;  in  other  words,  the  two 
motions  completely  annihilate  each  other. 

If  the  path  difference  has  another  value  than  in  the  two  cases 
considered  above,  there  results  from  the  interference  a  wave- 
motion  of  another  state  of  vibration  and  of  another  amplitude. 


For  example,  if  the  path  difference  amounts  to  J  \  or  £  \  (expressed 
generally:    [n  -f  J]  ^),  the  resulting  motion  is  shifted,  with  re- 


INTERFERENCE   OF  PLANE-POLARIZED   LIGHT 


spect  to  the  two  wave  systems  (provided  they  have  equal  am- 
plitude), by  i^,  as  may  easily  be  seen  from  Fig.  5.  Shifted 
just  as  much  in  the  opposite  direction  is  the  wave  arising  by 
interference,  if  the  two  wave  systems  have  a  path  difference  of 
J^,  |^,  .  .  .  (expressed  generally:  [n  +  }]^).  See  Fig.  6. 


Fig.  6. 

So  the  intensity  of  the  vibration  that  is  produced  by  the 
interference  of  two  plane-polarized  light  rays  with  parallel 
vibration  direction,  may  have  the  most  different  values  lying 
between  zero  and  the  sum  of  the  intensities  of  the  single  rays, 
according  to  their  difference  of  path;  but  the  vibration  plane  of 
the  resulting  light  ray  is  always  the  same  as  that  of  the  two 
interfering  rays. 

The  combination  of  two  plane-polarized  rays  transmitted 
along  the  same  line  leads  to  a  different  result,  on  the  other  hand,  if 
their  vibration  planes  form  an  angle.  Let 
us  consider,  for  example,  two  vibrations  of 
equal  amplitude  whose  vibration  directions 
form  a  right  angle  with  each  other;  and  let 
their  path  difference  be  J  ^,  or  expressed 
generally,  (n  +  J)  L  Then,  if  the  two  rays 
be  perpendicular  to  the  plane  of  Fig.  7  at 
the  point  a,  an  ether  particle  at  a  will  in 
virtue  of  the  one  motion  be  moved  in  the  Flg'  7' 

direction  aa.^  in  virtue  of  the  other,  toward  as.  At  the  instant 
when,  of  the  .former  motion,  one-fourth  of  its  vibration  period 
has  elapsed,  if  this  motion  alone  had  acted  the  particle  would 
be  at  a'  in  virtue  of  the  second  motion,  which  is  retarded 


(20  INTERFERENCE   OF  PLANE-POLARIZED   LIGHT 

JA  as  compared  with  the  first,  the  particle  would  be  just  about 
to  leave  its  position  of  rest, — i.e.  its  displacement  in  the  direction 
aa3  would  be  zero.  Accordingly  al  is  the  position  of  the  particle 
at  that  instant.  At  a  somewhat  later  instant  the  particle  would 
in  virtue  of  the  first  motion  have  been  moved  a  certain  distance, 
oijCty,  downward;  but  the  second  motion  has  acted  at  the  same 
time,  and  this,  alone,  would  have  moved  it  from  a  to  ax; 
according  to  the  law  of  the  parallelogram  of  motions,  therefore, 
the  position  of  the  particle  is  now  ay  After  half  the  vibration 
period  has  elapsed  the  particle  would  be  in  consequence  of  the 
first  motion  at  a,  in  virtue  of  the  second  at  a3;  the  first  elonga- 
tion is  zero;  so  at  this  instant  the  particle  reaches  «3.  And  so 
on.  The  path  of  the  ether  particle  will  in  the  case  before  us  be 
exactly  that  described  by  the  end  of  a  pendulum,  if  by  a  sudden 
push  one  moves  it  from  its  position  of  rest  and  then,  at  the 
instant  when  it  attains  its  greatest  distance  (at  al  in  Fig.  7) 
from  that  position,  gives  it  a  push  of  equal  strength  in  the  direc- 
tion a«3,  i.e.  tangentially :  its  path  is  a  circle.  Thus,  by  the 
interference  of  two  light  rays  plane-polarized  perpendicularly  to 
each  other  there  arises  light  vibrating  in  a  circle;  and  this  light 
has  the  same  vibration  period  as  had  the  two  single  rays,  since  a 
quarter  of  the  circular  path  has  been  passed  over  in  a  quarter  of 
the  vibration  period.  Such  light,  known  as  circularly  polarized 
light,  must  naturally  possess  different  properties  from  ordinary 
light.  It  agrees  with  ordinary  light  only  in  this :  that  it  exhibits 
no  one-sidedness  with  respect  to  a  plane  passing  through  the 
transmission  line  of  the  ray. 

In  the  foregoing  case,  if  the  two  interfering  rays  have  not 
equal  amplitude,  or  if  their  difference  of  path  is  other  than 
J^,  or  finally  if  the  angle  between  their  respective  vibration 
directions  is  not  90°,  then  in  general  there  results  a  motion  of 
the  ether  particles  in  an  ellipse;  and  the  ray  arising  by  the 
combination  is  said  to  be  elliptically  polarized.  In  different 
.planes  through  its  line  of  transmission  such  a  ray  must  behave 
.differently. 


PROPAGATION   OF   LIGHT  21 


OPTICALLY  ISOTROPIC  (SINGLY  REFRACTING) 

BODIES 

PROPAGATION  OF  LIGHT 

Up  to  this  point  a  ray  of  light  has  always  been  spoken  of  only 
as  a  straight  row  of  vibrating  ether  particles.  But  such  a  par- 
ticle belongs  not  only  to  one  row,  but  also  to  every  other  that 
we  obtain  if  we  imagine  straight  lines  as  drawn  from  the  par- 
ticle in  question  to  those  adjacent  to  it  in  all  directions.  Con- 
sequently, by  a  displacement  of  the  first  particle  the  equilibrium 
must  be  disturbed  and  a  force  awakened  in  all  these  rows; 
that  is,  the  light-motion  will  be  propagated  in  all  directions  in 
the  ether. 

The  manner  of  this  propagation  will  depend  on  whether  vibra- 
tory motions  of  any  one  kind  are  transmitted  along  different 
directions  with  equal,  or  with  different,  velocity;  in  other  words, 
it  will  depend  on  whether  the  ether  is  an  isotropic  medium 
(see  p.  3)  or  one  that  is  anisotropic  (heterotropic)  — one  in 
which  the  transmission  velocity  of  a  wave-motion  in  different 
directions  is  different. 

The  ether  is  isotropic  in  empty  space,  in  isotropic  (amor- 
phous) bodies  (see  p.  3),  and  in  the  crystals  of  one  definite  group. 
In  such  homogeneous  bodies  as  these  the  light  of  a  definite 
color  is  accordingly  transmitted  not  only  on  every  section  of  one 
of  the  above-mentioned  straight  lines,  — and  moreover  along 
the  two  opposite  directions  of  every  section,  —  but  also  on  all 
straight  lines,  of  whatever  direction,  with  equal  velocity.  Such 
bodies  are  said  to  be  optically  isotropic.  The  luminiferous 
ether  contained  in  them  differs  from  the  ether  of  space  in  this 
particular:  it  transmits  light  of  a  different  color  with  a  differ- 
ent velocity,  but  likewise  in  all  directions  with  equal  velocity. 
So  the  velocity  of  light  in  optically  isotropic  bodies  depends  not 
only  on  the  nature  of  the  body,  but  also  on  the  period  of  the 
light-vibrations. 


22 


OPTICALLY   ISOTROPIC   BODIES 


If  at  any  point  in  such  a  medium  there  begins  light-motion 
of  a  definite  vibration  period,  then,  since  the  motion  is  trans- 
mitted in  all  directions  equally  fast,  at  the  end  of  one  whole 
vibration  period,  T,  all  points  standing  at  a  distance  of  one 
wave  length  X  from  the  first  point,  i.e.  all  points  on  a  spherical 
surface  of  the  radius  X,  will  begin  their  motion  simultaneously. 
After  the  time  zT  the  same  will  be  done  by  all  the  points  of  a 
spherical  surface  having  the  radius  2^,  while  at  the  same 
instant  those  of  the  surface  first  mentioned  will  begin  their 
second  vibration.  And  so  on.  Just  as  a  row  of  ether  particles 
is  divided  by  the  vibratory  motion  into  a  number  of  equal  wave 
lengths,  so  does  the  ether  surrounding  the  starting-point  of  the 
light-motion  become  divided  into  a  number  of  spherical  shells 
standing  at  the  distance  \  from  one  another;  and  in  these  shells 
all  points  lying  at  equal  distance  from  the  boundary  of  two 
shells,  on  the  same  side,  are  in  the  same  state  of  vibration.  If 
from  the  starting-point  of  the  motion  we  lay  off  in  all  directions 
lengths  proportional  to  the  wave  length,  their  extremities  form  a 
surface  called  the  wave-surface  of  the  motion  emanating  from 

the  first  point.     This  surface 
p  contains    all    ether    particles 

that  begin  their  motion  simul- 
taneously. Thus,  in  an  iso- 
tropic  body,  as  glass,  a  light- 
motion  which  has  emanated 
from  the  point  C  (Fig.  8)  will 
have  arrived  after  the  definite 
time  T  at  the  surface  of  a 
sphere,  SS'.  At  that  instant 
every  point  of  this  surface, 
as  P,  begins  its  motion;  and 


Fig.  8. 


since  P,  as  also  the  point  A,  lies  at  once  on  all  possible  rows  of 
points,  i.e.  rows  of  ether  particles,  diverging  from  it,  it  must  dis- 
turb the  equilibrium  in  all  these  rows;  in  other  words,  a  like  wave- 
motion  must  proceed  from  the  disturbed  point  of  SS'  in  all 


PROPAGATION   OF   LIGHT  23 

directions.  Since,  therefore,  every  disturbed  point  of  such  a 
system  itself  thus  becomes  the  center  of  a  new  wave-surface, 
then  at  the  point  B  of  a  spherical  surface  ss',  whose  points 
later  begin  their  motion,  there  will  arrive  light-motion  not 
only  from  A,  but  also  from  every  other  point  of  the  first 
spherical  surface  SS'.  In  order  to  judge  of  the  effect  that  all 
these  motions  produce  on  the  motion  of  the  point  B,  one  must 
take  into  account  the  distance,  from  B,  of  the  points  whence  the 
motions  proceed.  If  we  observe  the  sphere  SS'  from  B  and 
imagine  circles  of  various  diameters  as  drawn .  on  its  surface 
about  the  point  A  (like  parallels  of  latitude  about  the  north  or 
south  pole  of  the  earth),  then  obviously  all  the  points  of  such 
a  circle  stand  equally  distant  from  B  [B  is  supposed  to  lie 
on  CA  prolonged];  the  several  circles,  on  the  other  hand,  lie 
at  different  distances  from  B.  If  in  addition  to  every  circle  we 
further  imagine  that  one  as  constructed  whose  distance  from  B 
is  greater  by  exactly  J^,  the  two  motions  that,  emanating  re- 
spectively from  a  point  of  the  one  circle  and  from  the  corre- 
sponding point  of  the  other,  come  together  at  B  will  by  their 
interference  totally  annihilate  each  other,  and  produce  no 
motion  at  all  at  B.  Now  if  one  compares  the  effect  on  the 
point  B  of  all  these  circular  zones  of  the  spherical  surface  SS', 
taking  their  area  into  account,  it  is  found,  as  the  aggregate 
result,  that  the  motions  coming  to  B  from  all  the  different  parts 
of  the  surface  are  entirely  destroyed  by  the  motions  coming 
from  other  parts;  with  the  exception  of  the  motion  emanating 
from  A.  Therefore  B  is  reached  only  by  that  vibratory  motion 
that  emanated  from  A ;  this,  alone,  sets  B  in  motion.  Since  the 
same  consideration  applies  to  every  point,  the  result  is  this: 
of  the  motion  that,  emanating  from  A,  must  in  a  certain  time 
have  been  transmitted  as  far  as  the  spherical  surface  ss't 
only  one  portion  can  exercise  an  effect, — namely,  the  portion 
that  comes  to  B;  and  in  like  manner,  from  P  motion  is  commu- 
nicated only  to  p  [on  CP  prolonged],  instead  of  in  all  directions, 
from  P'  only  to  p',  and  so  on. 


OPTICALLY  ISOTROPIC  BODIES 


The  points  at  which  the  motion  from  C  gradually  arrives 
thus  lie  on  a  straight  line;  that  is,  the  propagation  of  light  as  a 
wave-motion  of  the  ether  must  be  rectilinear. 

From  the  surface  SS',  to  which  the  light  has  been  propa- 
gated in  a  certain  time,  we  obtain  for  a  later  instant  the  cor- 
responding surface,  ss',  if  about  the  single  points  of  the  former 
surface  we  construct  wave-surfaces  with  the  radius  correspond- 
ing to  the  intervening  time,  and  find  the  surface  that  envelopes 
all  these  wave-surfaces.  (See  Tig.  9.)  If  we  connect  the 

point  of  tangency  of  each 
of  these  wave-surfaces  and 
the  enveloping  surface  with 
its  center,  e.g.  the  point  of 
tangency  p  with  P,  then  for 
each  singleluHace  we  obtain 
the  corresponding  ray.  This 
construction  (called,  after  its 
inventor,  Huygens's  con- 
struction) will  farther  on  be 
repeatedly  employed  to  find 
the  wave-surfaces,  and  there- 
by the  positions,  of  rays  — 
the  straight  lines  (AB,  Pp, 
P'p',  in  Fig.  9)  connecting 
the  centers  of  the  several  wave-surfaces  with  the  points  at  which 
these  surfaces  are  touched  by  the  enveloping  surface.  The 
surface  whose  radii  are  the  several  rays  is  therefore  called  also 
the  ray-surface. 

Now  if  we  consider  a  light  ray  as  a  row  of  transversely 
vibrating  ether  particles,  the  vibrations  for  any  ray  Cp  obvi- 
ously take  place  in  the  plane  (the  so-called  "  wave-plane  ") 
tangent  to  the  ray-surface  at  the  point  p\  further,  if  we  imagine 
a  bundle  of  parallel  rays  as  simultaneously  transmitted  along 
the  direction  Cp,  then  at  a  certain  instant  the  tangential  plane 
to  the  ray-surface  at  the  point  p  forms  their  "  front  ",  where- 


Fig.  9. 


PROPAGATION   OF  LIGHT  25 

fore  it  is  spoken  of  also  as  the  ray-front.  The  ray  that  be- 
longs to  a  given  ray-front  is  according  to  this  the  straight  line 
between  the  center  of  the  ray-surface  and  the  point  at  which  the 
ray-front  touches  the  ray-surface.  In  the  most  simple  case,  the 
one  before  us,  in  which  the  latter  has  the  form  of  a  sphere,  the 
ray  obviously  coincides  with  the  normal  to  the  ray-front  (desig- 
nated more  briefly  as  front-normal). 

If  a  medium  is  not  homogeneous  (heterogeneous),  the  ether  in 
different  parts  of  it  will  be  differently  constituted,  and  a  light 
ray  transmitted  in  the  medium  will  therefore  pass,  as  it  were, 
through  divers  media;  but  we  may  regard  such  a  body  as 
composed  of  divers  homogeneous  parts,  and  reduce  this  case 
to  that  of  the  homogeneous  body  if  we  know  what  change 
a  wave-motion  suffers  in  passing  from  one  medium  into 
another. 

If  we  imagine  an  ether  particle  at  the  boundary  of  two  bodies 
in  which  the  ether  is  differently  constituted  and  in  which  there- 
fore the  transmission  velocity  of  light  is  different,  this  particle  be- 
longs both  to  the  ether  of  the  one  body  and  to  that  of  the  other 
body;  consequently,  if  a  light-motion  reaches  such  a  particle  and 
sets  it  in  vibration,  the  equilibrium  must  thereby  be  disturbed  in 
all  the  rows  of  ether  particles  that  connect  the  particle  in  ques- 
tion with  the  adjacent  ones  of  the  first  medium,  — and  dis- 
turbed likewise  in  all  the  rows  of  the  second  medium  to  which 
the  considered  ether  particle  at  the  boundary  belongs.  Light- 
motion  must  therefore  be  propagated  backward  from  that  par- 
ticle into  the  first  medium,  but  at  the  same  time  also  penetrate 
the  second.*  A  ray  of  light  arriving  at  the  boundary  of  two 

*  That  the  motion  penetrates,  appears  to  be  the  case  only  with  transparent 
bodies;  yet,  since  likewise  the  apparently  opaque  exhibit  a  weakening  of  the 
reflected  light,  with  them  too  a  part  of  the  incident  light  must  have  penetrated. 
In  fact,  the  difference  consists  only  in  this:  that  in  transparent  bodies  the  light 
rays  can  penetrate  to  a  greater  depth  without  becoming  perceptibly  weakened 
(absorbed^,  but  with  the  so-called  "  opaque  "  bodies  only  to  a  slight  depth.  This 
weakening  (absorption)  of  the  light  consists  in  a  transference  of  the  motion  from 
the  ether  particles  to  the  smallest  parts  of  the  body  itself,  and  therefore  in  an 
apparent  loss  of  the  motion,  as  light. 


26  OPTICALLY   ISOTROPIC   BODIES 

media  must,  according  to  this,  become  divided  into  a  ray  that  is 
thrown  back  (reflected)  and  one  that  penetrates  the  second 
medium.  As  will  be  found  farther  on,  the  latter  ray  is  deflected 
(refracted)  from  its  original  direction. 

REFLECTION  OF  LIGHT 

The  law  according  to  which  a  ray  of  light  reflected  at  the 
boundary  of  two  media  is  transmitted  backward  into  the  first, 
can  easily  be  deduced  by  means  of  Huygens's  construction, 
explained  in  the  last  section.  This,  in  the  following,  will  again 
be  done  first  for  the  simplest  case,  that  of  two  optically  isotropic 
media.  If  upon  a  boundary-surface  of  the  same,  and  at  any 
angle,  there  fall  parallel  rays,  i.e.  rays  coming  from  a  source  of 
light  so  distant  that  a  number  of  mutually  adjacent  ones  may 
be  regarded  as  exactly  parallel  and  the  corresponding  area  of 
the  ray-surface  (a  sphere  with  infinite  radius)  as  absolutely  plane, 
then  the  area  of  the  separating  surface  on  which  they  impinge, 
since  it  is  exceedingly  small,  must  also  be  regarded  as  a  plane. 
If  MN  (Fig.  10)  be  the  intersection  of  this  plane  (perpen- 


A  B 

Fig.  10. 


dicular  to  that  of  the  figure)  with  the  plane  of  the  figure,  and 
PV-PVP±  three  rays  lying  equidistant  from  one  another  in  tfye 
plane  of  the  figure',  then  the  plane  perpendicular  to  these  rays 
through  ADE  is  their  ray-front  at  the  instant  when  the  first  ray 
strikes  the  boundary-surface.  During  the  time  that  elapses 


REFLECTION   OF  LIGHT  2/ 

before  also  the  last,  P3,  has  reached  the  boundary-surface  (at 
C),  the  reflected  first  ray  is  transmitted  from  A  just  as  far 
into  the  upper  medium;  i.e.  to  some  point  on  the  upper  half 
of  the  sphere  we  have  to  construct,  as  its  ray-surface,  about 
A  with  the  radius  EC.  The  ray  P2,  after  half  the  time  in- 
terval has  elapsed,  strikes  the  boundary-surface  at  B,  and 
during  the  second  half  of  this  time  it  is  transmitted  backward 
into  the  upper  medium  to  some  point  of  the  sphere  to  be 
constructed  about  B  with  the  radius  GC  =  J  EC.  Finally, 
the  ray  P3  reaches  the  boundary-surface  exactly  at  the  end 
of  the  same  time  interval;  so  the  ray-surface  of  the  reflected 
ray  belonging  to  it  is  a  sphere  constructed  about  C  with  the 
radius  zero,  — in  other  words,  the  point  C  itself.  The  com- 
mon ray  front  after  the  reflection  is  the  plane  tangent  to  the 
several  ray-surfaces;  and  since  obviously  by  this  plane  all 
spheres  whose  centers  lie  on  the  straight  line  MN  must  be 
touched  at  points  falling  in  the  plane  of  the  figure,  we  obtain 
the  intersection  of  the  reflected  ray-front  with  the  plane  of  the 
figure  if  we  draw  from  C  the  tangent  CHF.  The  reflected 
rays  are  thus  AF  (the  reflected  ray  Pt)  and  BH  (the  reflected 
ray  P2).  Now  since  according  to  construction  AF  =  EC,  and 
since  the  two  right-angled  triangles  ACF  and  ACE  have  their 
hypothenuse  in  common,  the  angle  CAF,  which  the  reflected 
ray  forms  with  MN,  is  equal  to  the  angle  ACE,  — i.e.  to  the 
angle  included  between  MN  and  the  incident  light  rays.  The 
same  applies  also  to  the  ray  BH.  In  other  words,  the  rays, 
parallel  before  the  reflection,  are  parallel  likewise  after  the  reflec- 
tion; and  their  common  ray- front,  CHF,  forms  the  same  angle 
with  MN  as  did  the  original  ray-front,  ADE,  but  lies  reversed  in 
its  relation  to  the  normal,  AL,  to  the  boundary-surface.  If  we 
call  this  normal  line  the  axis  of  incidence  and  the  angle  that  any 
incident  fay  forms  with  the  same  its  angle  of  incidence  (**),  des- 
ignating further  the  plane  through  the  incident  ray  and  the  axis 
of  incidence  (i.e.  the  plane  of  the  figure  in  Fig.  10)  as  the  plane 
of  incidence,  and  finally  the  angle  included  by  the  reflected  ray 


28 


OPTICALLY   ISOTROPIC   BODIES 


with  the  axis  of  incidence  as  the  angle  of  reflection,  — then  the 
law  of  the  reflection  of  light  reads: 

"  The  reflected  ray  lies  in  the  plane  of  incidence,  and  the 
angle  of  reflection  is  equal  to  the  angle  of  incidence." 

If  we  suppose  a  reflecting  plane  MN  (Fig.  n)  to  be  struck 
not  by  parallel  but  by  diverging  light  rays,  which  proceed  from 

a  luminous  point,  as  A,  situ- 
ated at  only  a  short  distance 
from  it,  then,  reflected  ac- 
cording to  the  law  just 
stated,  the  rays  will  diverge 
exactly  as  though  they  came 
from  a  point  Av  lying  on 
the  normal  to  MN,  or  to 
MN  prolonged,  just  as  far 
on  the  other  side  of  MN 
as  A  on  this  side.  The 
same  applies  to  every  other 

Fig.  lx.  point  of  a  luminous  body. 

Since  our  vision  is  so  or- 
ganized that  we  locate  the  image  of  an  object  wherever  the 
divergent  rays  entering  our  eye  converge  or  seem  to  converge, 
an  eye  looking  toward  the  reflecting  surface  in  the  direction 
from  E  will  see  A  in  the  position  Alf  B  in  the  position  Bv 
and  intermediate  points  of  the  object  AB  between  Al  and 
Bl — thus  an  image  of  the  object  at  AJ$r  The  plane  MN 
is  a  "  mirror-plane ",  or  plane  of  symmetry,  of  the  construc- 
tion shown  in  Fig.  n;  that  is,  the  image  AlBl  lies  equal  and 
opposite  to  AB  with  reference  to  that  plane;  we  therefore  say 
that  AB  and  AJS^  are  "  symmetrical  with  reference  to  MN  ". 
In  crystallography  the  law  of  the  reflection  of  light  finds  an 
important  application;  namely,  for  determining  the  angles  which 
the  plane  faces  of  crystals  form  with  one  another.  The  instru- 
ment employed  for  this  purpose  is  called  the  reflection  goni- 
ometer, and  its  principle  is  as  follows :  — 


REFLECTION   OF  LIGHT  2$ 

The  graduated  circle  C  (Fig.  12)  has  a  rotatable  axis  rigidly 
connected  with -the  index  or  vernier  F,  so  that  by  means  of  V 
one  may  read  off  on  the  graduations  of  the  circle  an  angle 
through  which  the  axis  has  been  rotated.  The  axis  projects 
in  front  of  the  circle,  where  it  carries  the  crystal,  this  being  so 


Fig.  12. 


fastened  that  the  intersection-edge  of  the  two  faces  whose  incli- 
nation is  to  be  measured  is  centered  (i.e.  in  its  prolongation 
strikes  the  center  of  the  circle)  and  adjusted  (i.e.  stands  normal 
to  the  plane  of  the  circle).  If  of  the  two  crystal  faces  desig- 
nated by  a  and  b  the  former  be  then  in  such  a  position  that  to 
the  eye  at  e  the  virtual  image  of  an  object  lying  in  the  direction 
co  appears  in  the  direction  eco',  and  if  we  subsequently  rotate 
the  axis,  and  with  it  the  crystal,  —  whereat  the  edge  a/b  of  course 
remains  motionless,  —  until  b  has  arrived  at  a  position  parallel 
to  that  previously  occupied  by  a,  the  crystal  will  now  have 
the  position  represented  by  the  dotted  outline,  and  we  shall 
now  see  the  reflected  image  of  o,  thrown  back  from  the  second 
face  in  the  same  direction,  ce,  in  which  it  was  previously 
reflected  from  the  first.  Hence  it  is  seen  from  Fig.  12  that 
the  angle  we  must  rotate  is  the  angle  at  which  the  two 


30  OPTICALLY   ISOTROPIC   BODIES 

planes  a  and  b  intersect  each  other,  and  that  accordingly 
the  exterior  angle*  of  the  crystal  is  determined  when  this 
rotation  angle  is  read  off  on  the  circle. f 

REFRACTION  OF  LIGHT 

As  was  mentioned  on  page  25,  a  wave-motion  arriving  at  the 
boundary  of  two  media  is  in  general  divided  there  into  a 
reflected  motion  and  a  penetrating  motion.  The  direction  of 
the  wave-ray  penetrating  the  second  medium  is,  as  will  now  be 
shown,  different  from  that  in  the  first,  provided  the  transmission 
velocity  of  the  ray  is  different  in  the  two  media. 

Let  us  consider  first  the  case  of  a  light  ray  transmitted  in 
the  second  medium  with  less  velocity  than  in  the  first.  If 
Plf  P2,  P3,  P4  (Fig.  13)  be  rays  lying  so  close  together  and  derived 
from  a  source  of  light  so  remote  that  we  may  regard  them  as 
exactly  parallel  and  the  portion,  CD,  of  the  ray- front  lying 
between  them  as  a  plane,  then  at  a  certain  instant  the  ray  Pl 
will  strike  the  boundary-plane,  MN,  at  D.  From  thence 
onward  the  penetrating  wave-motion  is  transmitted  in  the 
second  medium,  but  with  less  velocity.  If  this  velocity  be,  for 
example,  only  half  that  in  the  first  medium,  then  after  the  time 
t  has  elapsed,  which  the  ray  P4  requires  to  travel  the  distance 

*This  is  called  the  "normal  angle"  (because  it  is  equal  to  that  included 
between  the  normals  to  the  faces  a  and  6),  or  simply  the  "angle",  of  the  faces  a 
and  b.  Its  supplement  is  the  interior,  or  actual  "  interfacial ",  angle  —  the  angle 
the  two  faces  a  and  b  include  within  the  crystal. 

t  [Particulars  as  to  the  arrangement  and  use  of  the  reflection  goniometer,  as 
also  of  other  instruments,  apparatus,  and  various  appliances  employed  in  the  investi- 
gation of  crystals,  will  be  found,  illustrated  with  numerous  figures,  in  Part  III  of 
the  author's  "Physikalische  Krystallographie".  Leipzig;  3rd  ed.  1895,  4th  ed. 
1905.  See  also  the  trade  catalogues  (cf.  Appendix)  and  C.  Leiss:  "  Die  optischen 
Instrumente  der  Firma  Fuess".  Leipzig,  1899.  In  English  many  of  the  same 
are  more  briefly  described  in  various  works,  especially  in  E.  S.  Dana:  "A  Text- 
book of  Mineralogy",  New  York,  1904,  and  in  A.  J.  Moses:  "  The  Characters  of 
Crystals  ",  New  York,  1899.  Cf .  also  footnote  f,  p.  80.  A  very  comprehensive 
and  profusely  illustrated  French  work  is  that  of  Duparc  and  Pearce  (L.  Duparc  et 
F.  Pearce:  "  Traite*  de  technique  mineralogique  et  petrographique.  ire  partie: 
Les  methodesoptiques".  Leipzig,  1907.  Additional  literature  is  mentioned  ibid.] 


REFRACTION   OF   LIGHT  31 

CA,  P1  will  have  penetrated  into  the  second  medium  a  distance 
amounting  only  to  the  half  o.  CA.  Pl  will  then  have  reached 
some  point  of  the  hemisphere  described  about  D  with  the  radius 


Db1  =  J  CA.  The  second  ray,  P2,  strikes  the  boundary-surface 
MN  somewhat  later,  namely,  at  the  instant  when  P4  has  arrived 
at  the  point  c2;  up  to  the  end  of  the  time  t  the  ray  P4  moves  in 
the  first  medium  a  distance  c2A  forward;  the  ray  P2,  during  the 
same  time  in  the  second  medium,  can  travel  only  half  that  dis- 
tance. This  ray  P2,  therefore,  at  the  end  of  the  time  t,  will  be 
at  a  point  of  the  hemisphere  constructed  about  p2  with  the 
radius  p2b2  =  J  c2A.  In  the  same  way,  after  the  time  /  has 
elapsed  the  ray-surface  of  the  ray  transmitted  from  p3  into  the 
second  medium  will  be  a  hemisphere  about  p3  with  the  radius 
p3b3  =  J  c3A.  Finally,  that  of  the  ray  P4  at  the  same  instant 
will  be  the  point  A  itself.  The  common  front  of  all  rays  lying 
between  Px  and  P4  in  the  second  medium  is  the  surface  tangent 
to  all  their  respective  ray-surfaces  in  that  medium;  namely,  the 
plane  passing  through  AB  perpendicular  to  the  plane  of  Fig.  13. 
We  should  also  have  found  this  plane  had  we  constructed  the  ray- 
surface  only  of  one  ray,  as  P1?  and  drawn  the  tangent  to  it  from 
A.  So  the  plane  ray-front  CD  in  the  first  medium  is  in  the 
second  still  plane,  but  has  a  different  direction.  The  latter  is 
naturally  true  also  of  the  rays — the  straight  lines  between  the 
points  D,  p2,  p3,  etc.,  and  the  points  of  tangency  bv  b2,  b3,  etc. 


32  OPTICALLY   ISOTROPIC   BODIES 

From  Huygens's  construction  we  accordingly  see  in  the 
first  place  that,  although  the  wave-motion  penetrating  the  sec- 
ond medium  remains  in  the  plane  containing  the  incident  ray 
and  the  axis  of  incidence  (GF,  Fig.  14),  i.e.  in  the  plane  of  inci- 
dence, yet  it  is  deflected  from  its  direction,  or  refracted.  From 
this  construction,  then,  we  may  also  deduce  the  law  after  which 
the  refraction  takes  place. 


Fig.  14. 

According  to  construction  the  lengths  CA  and  DB  are  to 
each  other  as  the  transmission  velocity  of  light  in  the  first  medium 
is  to  that  in  the  second  medium  (in  our  example  as  2  :  i);  if  we 
call  the  former  velocity  v  and  the  latter  v'j  then 


BD 


But  the  angle  of  incidence  i  =  P^DG  =  A  DC,  and  the  angle 
BDF,  which  the  refracted  ray  forms  with  the  axis  of  incidence 
and  which  is  called  the  angle  of  refraction  (r),  =  /  DAB.  In 
the  two  right-angled  triangles  A  CD  and  ABD  we  have 


^  =  sin  ADC  =  sin  *, 
AD 

— -  =  sin  BAD  =  sin  r. 
AD 

Dividing  the  first  expression  by  the  second  gives 
AC  _  sin  im 
BD  ~~  sin  r' 


REFRACTION   OF  LIGHT  33 

and  since  AC  and  BD  are  proportional  to  the  transmission 
velocities, 

sin  i  =  v_ 
sin  r      vf 

According  to  this,  the  angle  of  incidence  may  have  any  size, 

but  ITS  SINE  MUST  ALWAYS  BE  TO  THE  SINE  OF  THE  ANGLE  OF  RE- 
FRACTION AS  THE  TRANSMISSION  VELOCITY  IN  THE  FIRST  IS  TO 

THAT  IN  THE  SECOND  MEDIUM.     This  ratio  —  is  called  the  re- 

v' 

fractive  index*  (index  of  refraction,  refraction  quotient,  refrac- 
tion exponent) ,  and  in  the  following  is  denoted  by  n.  If,  therefore, 
by  determining  the  direction  of  any  incident  ray  and  of  the 
corresponding  refracted  ray,  n  has  once  been  found  for  two 
optically  isotropic  media,  then  for  every  ray  of  light  falling  in 
a  different  direction  upon  the  boundary-surface  of  the  same  two 
media  it  permits  the  direction  of  the  corresponding  refracted 
ray  to  be  calculated. 

In  the  cases  hitherto  considered  the  lesser  light  velocity  was 
that  in  the  second  medium:  v  was  greater  than  vf  (v  >  vf)  and 

accordingly  the  refractive  index  —  >  i.     In  this  case,  for  every 

angle  of  incidence,  the  angle  of  refraction  is  smaller  than  the 
angle  of  incidence:  the  ray  is  refracted  toward  the  axis  of 
incidence. 

When  on  the  other  hand  the  light  ray  is  transmitted  in  the 
second  medium  with  the  greater  velocity,  i.e.  when  v'  >  v  and 
consequently  n  <  i,  then,  as  follows  from  the  construction  in 

*  The  term  "  refractive  index"  of  an  optically  isotropic  solid  usually  refers 
to  the  refraction  from  air  into  the  solid,  to  the  ratio  therefore  of  the  light  velocity 
in  air  to  that  in  the  solid;  and  this  ratio,  since  in  nearly  all  solid  bodies  light  travels 
more  slowly  than  in  air,  is  in  most  cases  greater  than  unity.  Since  for  light  rays 
of  any  one  color  the  definite  numerical  value  of  n  is  always  the  same  with  one  and 
the  same  substance,  then,  under  circumstances  that  do  not  admit  of  the  determi- 
nation of  other  properties  (e.g.  in  the  case  of  gems),  this  value  may  be  employed 
for  identifying  the  substance. 


34 


OPTICALLY   ISOTROPIC    BODIES 


Fig.  15,  — which,  being  wholly  analogous  to    those  preceding, 
needs  no  special  explanation,  —  the  ray  is  refracted  away  from 

the  axis  of  incidence.  Since 
accordingly  the  angle  of  re- 
fraction is  here  greater  than 
the  angle  of  incidence,  there 
must  be  a  value  of  i  for 
which  r  =  90°.  This  is  the 
case  when  sin  i  =  n;  for 
sin  r  must  then  be  unity. 
That  ray,  therefore,  whose 
angle  of  incidence  is  of  such 
a  size  that  its  sine  is  exactly 
equal  to  the  refraction  ex- 
ponent, is  so  refracted  that  it 
is  transmitted  parallel  to  the  boundary-surface,  and  thus  it  does 
not  penetrate  the  second  medium.  Just  as  little,  according  to 
the  law  of  refraction,  can  a  ray  penetrate  whose  angle  of  inci- 


dence is  still  greater;  for  then 


sin 


>  i. 


rp,  .         ,       sin  i  . 
This  value  is 


the  value  for  the  sine  of  the  angle  of  refraction;  but  no  angle 
exists  whose  sine  is  greater  than  unity;  so  in  this  case  also  there 
is  no  angle  of  refraction. 

Figure  15  represents  the  refraction  at  the  boundary  of  one 
medium  with  a  second  (the  lower),  in  which  the  velocity  is 
double  that  in  the  upper  medium,  so  that  the  refractive  index 
is  J.  The  ray-front  in  the  second  medium  is  found,  in  a  simpli- 
fied manner,  as  the  tangential  plane  AB  to  the  wave-surface  of 
the  ray  Pl  in  the  lower  medium,  this  ray-surface  being  described 
about  D  with  the  radius  DB  =  2  AC.  Accordingly  DB  is  the 
direction  of  the  refracted  ray.  In  Fig.  16  the  angle  i  is  such 

CM    /""<  \ 
=  — —  J,  has  the  value  n  =  -;so  that^4Z>  = 
A.D  I  2 

2  AC.  The  wave-surface  of  the  ray  Pl  must  be  constructed 
about  D  with  the  radius  DA,  and  thus  it  passes  through  A\  the 


REFRACTION   OF   LIGHT 


35 


tangent  to  the  circle  from  the  point  A,  i.e.  the  front  of  the  re- 
fracted ray,  has  the  direction  AB;  consequently  the  ray  itself, 


Fig.  16. 

the  straight  line  from  the  point  of  tangency  to  the  center,  lies 
in  MN:  the  angle  of  refraction  is  90°,  as  was  already  taught  by 
the  above  calculation.  Figure  17,  finally,  shows  the  refraction 


in  the  case  of  a  still  greater  angle  of  incidence,  under  otherwise 
the  same  conditions.  The  wave-surface  of  the  ray  Pl  in  the 
second  medium  is  the  spherical  surface  described  about  D  with 


OPTICALLY   ISOTROPIC  BODIES 


the  radius  DB  =  2  AC;  to  this  surface  no  tangential  plane  may 
be  passed  from  A,  because  A  lies  within  it,  and  therefore  in  the 
second  medium  there  is  no  ray-front  and  no  refracted  ray. 

The  law  of  the  refraction  of  light  thus  teaches  (i)  that  a  light- 
motion  arriving  at  the  boundary  of  two  media  is  always  deflected 
from  its  direction;  (2)  that  the  motion,  falling  on  the  boundary  in 
whatever  direction,  always  penetrates  the  second  medium,  pro- 
vided its  transmission  velocity  in  this  medium  is  less  than  in  the 
first;  (3)  that,  on  the  other  hand,  in  the  reverse  case  an  entrance  of 
the  motion  can  take  place  only  when  the  angle  of  incidence  does 
not  exceed  a  certain  limit.  With  a  greater  angle  of  incidence  the 
incident  ray  has,  on  the  contrary,  no  longer  a  refracted  ray: 
there  no  longer  takes  place  a  division  of  the  motion  into  a  re- 
flected part  and  a  refracted  part: 
the  whole  motion  is  reflected. 
Therefore  this  phenomenon  is 
called  the  total  reflection  of  light; 
and  the  smallest  angle  of  inci- 
dence with  which  it  occurs  — 
whose  value,  according  to  the 
foregoing,  follows  directly  from 
the  refractive  index  —  is  termed 
the  critical  angle  of  total  reflec- 
tion. 

A  ray  of  light  falling  on  the 
plane    surface    of    a   transparent 
optically  isotropic  body  that  has  a  second  face  parallel  to  the 

first  (Fig.  1 8)  is  at  the  first  face  so  refracted  that  ^n-^=  ^-,and 

sin  r     v' 

a,t  the  second  so  that  - —  =  — .     For  these  values  are  the  ratios 
sin  e      v 

of  the  transmission  velocity  of  the  light  in  the  medium  from 
which  it  arrives  at  the  second  boundary,  to  that  in  the  medium 
into  which  it  emerges.     But  thence  follows 
sin  e  =  sin  i,    e  =  i: 


Fig.  18. 


REFRACTION   OF  LIGHT 


37 


that  is,  the  ray  continues  its  way  on  the  other  side  of  the  second 
boundary-surface  along  the  same  direction  in  which  it  fell  upon 
the  first.  So  a  plane-parallel  transparent  plate  introduced  into 
the  path  of  parallel  light  rays  does  not  change  their  direction. 
It  is  at  the  same  time  evident,  from  the  foregoing,  that  the 
refractive  index  for  passage  from  one  medium  into  a  second  is 
the  reciprocal  of  that  in  the  case  of  passage  from  the  second  into 
the  first;  for  example,  if  the  refractive  index  of  a  ray  passing 

from  air  into  glass  is  n}  that  for  the  exit  into  air  is  —  • 

Now  if  we  consider  the  case  of  a  light  ray  entering  and 
emerging  from  a  transparent  body  whose  faces  of  entrance  and 
emergence  are  not  parallel,  it  is  at  once  clear  that  the  emergent 
ray  cannot  then  be  parallel  to  the  in- 
cident ray.  If  Fig.  19  represent  the 
cross-section  of  such  a  body,  called  a 
prism,  bounded  by  the  two  planes  MN 
and  MO,  the  construction  shows  that 
a  light  ray,  LL',  passing  through  the 
prism  experiences  a  deviation.  We 
may  calculate  this  deviation  if  we  know 
a,  the  so-called  refracting  angle  of  the 
prism,  together  with  the  angle  of  in- 
cidence i  and  the  refractive  index  n. 
But  conversely,  with  the  aid  of  such 
a  prism  we  are  able,  by  measuring 
the  deviation  it  produces,  to  determine  the  refractive  index 
n  of  a  light  ray  passing  from  air  into  the  substance  of  the 
prism. 

The  solution  of  this  problem  assumes  an  especially  simple 
form  when  the  entering  and  the  emerging  ray  include  equal 
angles  with  the  two  faces  of  the  prism  respectively,  in  other 
words,  when  i  =  if  and  x  =  y;  because  then  (as  may  be 
proved  in  various  ways)  the  deviation,  d,  produced  by  the  prism 
assumes  its  minimum  value. 


OPTICALLY   ISOTROPIC   BODIES 


This  case  is  represented  in  Fig.  20,  where  ANOC  indicates  the 
path  of  the  ray.     Since  here  Z  MNO  =  Z  MON,  the  direction, 

NO,  of  the  ray  within  the  prism 
is  perpendicular  to  the  bisector 
of  the  prism-angle;  consequently 


Z  ONL'  =  - 

2 


If  Nm  and  Om 


Fig.  20. 


be  the  prolongations  of  the  two 
directions  of  the  ray  outside  the 
prism,  and  if  no  be  parallel  to 

NO,  then  Z  Omo  =  Z  Nmn  =  -  , 

2 

where  d  stands  for  the  total  devi- 
ation produced  by  the  prism;  i.e. 
for  the  supplement  of  the  angle 
AmC.  If  we  prolong  ON  to 
B,  the  angle  of  incidence  i  = 
ANL  =  ANB  +  BNL.  But 


=  ONm  =  Nmn  =  -,  and 


BNL 


a 


l   = 


a  + 


ONL'  =  — ;  consequently  the  whole  angle  of  incidence 
2 

,  and  for  the  refraction  from  air  into  the  prism  the 


a 


angle  of  refraction  r  =  ONL'  =  - .      Therefore  the  refractive 
index 


sin 


a  4- 


sin  i 
sin  r 


.    a 

sin- 

2 


Now  this  last  equation  makes  possible  the  most  usual  method 
of  determining  the  refractive  index  of  a  light  ray  passing 
from  air  into  a  transparent  solid.  For  this  purpose  one  has 
only  to  determine  the  refracting  angle  a  of  a  prism  composed  of 


REFRACTION    OF   LIGHT 


39 


the  solid  and  the  minimum  deviation  of  a  light  ray  passing 
through  the  prism.  The  former  of  these  quantities  is  determined 
by  means  of  the  reflection  goniometer  (see  p.  29) ;  the  latter 
likewise  by  means  of  a  graduated  circle  at  whose  center  the 
prism  is  so  adjusted  as  to  be  rotatable,  but  on  which  circle  one 
may  read  off,  besides  the  direction  of  the  ray  emerging  from  the 
prism,  that  of  the  entering  ray  as  well.  The  diagram  Fig.  21 


Fig.  21. 

may  serve  to  illustrate  this.  Here  P  is  the  center  of  the  circle, 
in  front  of  which  is  the  refracting  edge  of  the  prism,  and  AP  || 
to  BP  is  the  direction  of  the  incident  light,  PC  that  of  the  re- 
fracted ray;  the  angle  BPC,  to  be  read  on  the  graduated  circle, 
is  then  the  deviation  d.  One  rotates  the  prism  about  its  refracting 
edge,  i.e.  about  the  axis  of  the  circle,  until  d  has  its  minimum 
value  — until,  with  further  rotation,  the  ray  PC  would  move 
farther  away  from  PB.  When  this  value,  as  well  as  that  of  the 
refracting  angle  a,  has  been  determined,  the  refractive  index 
sought  is  found  directly  from  the  equation 


sin 


.    a 

sin- 

2 


*  For  details  of  the  practical  manipulations  in  such  a  measurement  cf.  foot- 
note f,  p-  30. 


4o 


OPTICALLY   ISOTROPIC    BODIES 


The  foregoing  method  of  determining  the  refractive  index  is 
the  most  accurate,  but  requires  a  carefully  ground  prism  of  a 
certain  size.  However,  it  is  also  possible,  with  the  aid  of  a  small 
plate  of  a  transparent  substance,  to  determine  the  refractive 
index  of  the  same;  and  this  may  be  done  by  two  different 
methods. 

The  first  of  these  methods  depends  on  the  law  of  total  reflec- 
tion (see  p.  36),  and  as  employed  by  Kohlrausch  consists  in 
measuring  the  critical  angle  of  total  reflection  in  the  following 
way:  In  a  glass  vessel  (Fig.  22)  a  liquid,  /,  is  contained  whose 


Fig.  22. 

refractive  index  must  be  larger  than  that  of  the  substance  to  be 
investigated.  (In  order  to  render  this  method  applicable  to  the 
largest  possible  number  of  substances  a  liquid  having  very 
great  "  refringency",  or  power  to  refract,  as  bromnaphthalin,  is 
chosen.)  In  front  the  otherwise  cylindrical  vessel  consists  of  a 
plane-parallel  glass  plate,  p,  before  which  and  directed  perpen- 
dicular to  which  is  a  small  telescope  focused  on  an  object  at 
infinite  distance,  hence  for  parallel  rays.  On  a  rotatable  sup- 
port plunging  into  the  liquid  from  above,  the  plate  of  the  sub- 
stance to  be  investigated  is  so  fastened  that  the  rotation  axis 
falls  in  its  (plane)  front  face,  which  is  illuminated  from  the  side 


REFRACTION   OF   LIGHT  4! 

by  diffused  light.  Among  the  light  rays  falling  on  this  face  at 
various  inclinations  are  such  as  have  a  direction  BA ;  —  and  let 
it  be  these  that  form  with  the  normal,  A  N,  an  angle  i,  the  critical 
angle  of  total  reflection.  If  one  rotates  the  plate  to  be  investi- 
gated into  exactly  the  position  indicated  in  Fig.  22,  this  position 
being  such  that  the  rays  reflected  parallel  to  AC  (CAN  =  BAN 
=  i)  converge  at  the  center  of  the  field  of  the  telescope,  then 
the  one-half  of  the  field  must  receive  rays  falling  on  the  plate  at 
angles  smaller  than  i,  which  rays  therefore  penetrate  in  part, 
being  only  in  part  reflected;  while  the  other  half,  on  the  other 
hand,  receives  such  rays  as  have  a  larger  angle  of  incidence  and 
which  are  therefore  reflected  with  their  full  intensity.  The 
latter  half  of  the  field  must  in  consequence  appear  brighter  than 
the  former  half  and  be  separated  from  it  by  a  vertical  boundary- 
line,  which  passes  exactly  through  the  center  of  the  field;  hence 
this  position  of  the  plate  may  be  found  by  adjusting  the  bound- 
ary-line on  the  center  of  the  field.  If,  then,  one  causes  diffused 
light  to  fall  from  the  other  side  of  the  vessel  and  rotates  the  plate 
until  that  boundary-line  again  appears  at  the  center  of  the  field, 
the  plate  now  has  the  position  indicated  in  dots,  and  the  direc- 
tion AB  has  taken  the  position  of  AC,  while  the  rays  correspond- 
ing to  the  critical  angle  of  total  reflection  fall  in  the  direction 
DA.  The  angle  the  plate  must  be  rotated  is  obviously  2ij  and 
hence,  by  reading  off  this  rotation  on  a  graduated  circle  and 
dividing  by  two,  one  obtains  the  critical  angle  of  total  reflection. 
But  according  to  page  33,  the  sine  of  this  angle  is  equal  to  the 
refractive  index  —  the  ratio  of  the  light  velocity  in  the  liquid, 
L,  to  the  light  velocity  in  the  plate,  i>;  therefore  we  have 

L  L 

sin  z  =  -  ,  whence  v  =  - — .  • 
i)  sin  i 

If  we  divide  each  member  of  the  last  equation  into  the  value  of 
the  light  velocity  in  air  —  let  it  be  V  —  we  obtain 

V       V   . 
—  =  —  sin  i. 
v      L 


42  OPTICALLY   ISOTROPIC   BODIES 

y 

But  — ,  the  ratio  between  light  velocity  in  air  and  in  liquid1,  is 
JL/ 

the  refractive  index  of  the  latter,  and  this  index  can  be  found  with 
the  aid  of  a  hollow  prism  filled  up  with  the  liquid,  by  the  method 
first  given,  of  minimum  deviation.  If  it  be  determined  once 
for  all,  for  the  liquid  in  question,  and  be  denoted  by  //,  then 

V 
JJL  sin  i  =  —  • 

v 
y 
But  the  quotient  —  ,  the  ratio  of  the  light  velocity  in  air  to  that  in 

the  plate,  is  the  required  refractive  index  of  the  latter,  which 
accordingly  is  found  by  multiplying  the  sine  of  the  measured 
critical  angle  of  total  reflection  by  the  refractive  index  of  the 
liquid  employed. 

Instead  of  a  liquid  we  may  use  a  transparent  optically 
isotropic  solid  having  higher  refringency  than  the  one  to  be 
investigated  (Wollaston's  method) ;  and  for  accuracy  of  the  deter- 
mination this  method  has  a  great  advantage.  For  the  refractive 
index  of  every  substance  varies  with  the  temperature,  and  in 
the  case  of  liquids  this  variation  is  very  considerable,  wherefore 
in  using  the  method  just  described  an  inavoidable  rise  in  tem- 
perature perceptibly  influences  the  result;  while  in  the  case  of 
solid  bodies  the  variations  are  much  less.  To  which  must  be 
added  that  it  is  possible  to  manufacture  homogeneous  glasses 
whose  refractive  index  amounts  to  1.92,  thus  being  greater  than 
that  of  most  other  substances.  By  reason  of  these  aijd  other 
advantages  the  refractometer  constructed  by  Abbe  and  Czapski 
is  especially  adapted  for  determining  the  refractive  indices  of 
crystals;  the  instrument  named  is  based  on  the  following  prin- 
ciple: — 

The  plate,  P  (Fig.  23),  of  the  body  to  be  investigated  is  laid 
with  its  plane  face  on  the  plane  surface  of  a  hemisphere  com- 
posed of  the  highly  refractive  glass,  and  by  means  of  a  very 
thin  plane-parallel  layer  of  a  highly  refractive  liquid  the  two 
planes  are  brought  into  perfect  contact.  Let  ON  be  the  normal 


REFRACTION   OF  LIGHT 


43 


to  the  separating  surface,  and  NOJ  (or  NOJ')  the  critical  angle 
of  total  reflection.  Then,  if  light  rays  fall  from  below  on  the 
right,  as  indicated  by  the  radial  lines  , 

in  the  figure,  falling  in  part  at  larger,  in 
part  at  smaller  angles,  the  former  rays 
are  totally  reflected,  while  those  falling 
at  the  smaller  angles  are  only  partially 
reflected.  (In  the  figure  this  is  indicated 
by  the  lines  being  dotted.)  Conse- 
quently, in  a  telescope  focused  for  infi- 
nite distance,  whose  axis  is  brought  into 
the  direction  OJ,  the  left  half  of  the  field 


Fig.  23. 


will  appear  bright  but  the  right  half  less  bright;  that  is,  a  verti- 
cal boundary  will  pass  through  the  center  of  the  field.  Since  of 
rays  that  are  parallel  to  a  radius,  which  must  accordingly  have 
passed  through  the  spherical  surface  perpendicularly,  no  devia- 
tion takes  place,  the  reading  of  the  position  of  the  telescope  on 
a  graduated  circle  whose  zero  point  corresponds  to  the  radius 
ON  gives  directly  the  critical  angle  of  total  reflection. 

Still  more  accurate  is  the  adjustment  of  the  boundary  if  one 
employs  a  modified  procedure,  for  which  it  is  requisite  that  the 
plate  laid  upon  the  hemisphere  permit  the  lateral  entrance  of 

light;  the  best  results  are  obtained  when 
the  plate  is  circular  and  bounded  by  a 
polished  cylindrical  surface,  so  that  its 
cross-section  has  the  form  shown  in  Fig. 
24.  (The  upper  side  may  be  rough-ground 
or  otherwise  constituted.)  Should  one, 
then,  cause  light  rays  to  fall  upon  the 
hemisphere  from  below  on  the  left  in 
directions  adjacent  to  7O,  the  rays  that 
fall  exactly  parallel  to  JO  would  be  transmitted  in  the  boundary 
plane, — parallel  to  OM  therefore,  — while  those  falling  within  the 
angle  JON  (in  the  figure  these  are  given  in  dots)  would  leave  the 
plate  through  the  side  face.  Hence,  if  we  imagine  this  optical 


44  OPTICALLY   ISOTROPIC    BODIES 

construction  as  reversed  (see  p.  14),  i>e.  the  rays  as  entering 
through  the  cylinder  face  and  on  the  right  from  above,  then  to  the 
incident  ray  OM  grazing  the  boundary-surface  there  belongs  the 
ray  transmitted  parallel  to  OJ;  while  to  the  rays  following  above 
OM  there  belong  those  represented  in  dots,  wherefore  in  the  tele- 
scope directed  along  JO  the  right  half  of  the  field  appears  illu- 
minated as  in  the  last  case.  (See  p.  43.)  Now  if  it  is  so  arranged 
that  from  below  the  plane  OM  not  the  least  light  can  fall,  the  left 
half  of  the  field  appears  absolutely  dark  and  the  boundary  there- 
fore much  sharper  than  before.  This  procedure,  called  that  of 
grazing  incidence,  can  of  course  be  employed  also  with  the 
Kohlrausch  total-reflectometer. 

Finally,  the  microscope  too  may  be  used  for  the  determina- 
tion of  refractive  indices.     Such  a  method  was  suggested  as 
early  as  the  eighteenth  century  by  the  Duke 
of  Chaulnes,  and  depends  on  the  principle 
that  the  focal   distance   of  a   microscope 
changes  when  a  transparent  plane-parallel 
plate  is  inserted  between  the  objective  and 
the  focus.     As  shown  in  Fig.  25,  the  rays 
i    appear  to  come  from  a  point  0',  while  in 
1  reality  they  proceed   from  a.      So  if  the 
p.   '2  microscope  has  been  accurately  focused  on 

any  object,  and  if  a  lamella  of  the  substance 
is  now  inserted  above  the  same,  then  in  order  to  again  see  the 
object  distinctly  one  must  alter  the  distance  of  the  microscope 
from  it  a  certain  amount;  by  means  of  graduations  on  the 
fine-adjustment  screw  of  the  microscope  this  amount  can  be  very 
accurately  measured.  Among  the  change,  d,  in  the  focal  distance 
of  the  objective,  the  thickness,  e,  of  the  inserted  lamella,  and  the 
refractive  index  n  of  the  latter  there  exists  the  relation 


n  = 


e  -d' 

from  this  equation,  after  e  and  d  have  been  measured,  the  refractive 
index  may  be  calculated. 


REFRACTION   OF   LIGHT  45 

Another  method  depending  on  the  total  reflection  of  light 
is  due  to  Becke.  If  in  the  field  of  the  microscope  two  trans- 
parent bodies  lie  contiguous  to  each  other,  the  refringency  of 
one  of  them  being  known,  and  if  the  cone  of  illumination  of  the 
microscope  is  narrowed  down  to  the  critical  angle  of  total 
reflection  of  the  more  refractive  body,  this  body  exhibits  a 
bright  border  if  we  adjust  the  microscope  on  the  upper  edge 
of  the  boundary-surface,  while  if  the  microscope  is  lowered  the 
bright  border  appears  in  the  medium  of  the  lesser  refringency. 
One  can  thus  demonstrate  even  very  small  differences  in  re- 
fringency, and,  with  the  aid  of  a  scale  of  substances  having 
known  refractive  indices,  determine  those  of  a  body  to  be 
investigated.* 

If  in  one  of  the  ways  mentioned  the  refractive  index  of  an 
optically  isotropic  body  is  determined  for  a  light  ray  of  a  certain 
color,  as  well  as  for  a  ray  of  another  color,  the  two  values  are 
found  to  differ.  In  accord  with  this,  it  is  seen  that  by  a  prism  a 
ray  of  white  light  is  split  up  into  rays  of  different  color,  which 
in  the  prism  have  all  experienced  unequal  deviation.  Since 
these  colors,  as  also  their  sequence  in  deviation,  are  in  general 
the  same  with  prisms  of  different  substances,  they  cannot  have 
been  created  by  the  prism,  but  must  have  preexisted  in  the 
incident  white  light.  It  must  accordingly  be  assumed  that 
so-called  white  light  is  composed  of  the  different  colors  which 
are  scattered  (dispersed)  by  the  refraction  into  a  spectrum.  In- 
deed, we  are  able  by  means  of  a  similar  prism  in  the  reverse 
position  to  unite  these  rays,  and  they  then  again  produce  in 
our  eye  the  impression  of  white.  The  color  spectrum  which  the 
white  light  yields  with  a  prism  has  a  greater  extent  when  the 
prism  has  a  greater  refracting  angle,  because  with  an  increase 
of  the  latter  there  is  an  increase  in  the  deviation  of  every  color 
and  consequently  in  the  distance  of  the  least  refrangible  ray  in 
the  spectrum  from  the  ray  most  strongly  refracted.  But  the  length 

*  Concerning  more  detailed  information  on  all  the  before-mentioned  methods 
and  apparatus  see  footnote  f,  p.  30. 


46  OPTICALLY   ISOTROPIC    BODIES 

of  the  spectrum  depends,  besides,  on  the  dispersive  power  of  the 
substance  composing  the  prism;  that  is,  on  the  power  the  sub- 
stance of  the  prism  possesses  to  scatter  the  different  colors. 

Empirically,  therefore,  we  learn  that  the  refractive  index 
varies  not  only  with  the  substance,  but  also  with  the  color  of  the 
light;  i.e.  with  the  vibration  period  and  hence  with  the  wave 
length.  Accurate  theoretical  deduction  of  the  transmission 
velocity  of  any  wave-motion  does  indeed  reveal  a  dependence  of 
the  same  on  the  wave  length  of  the  motion.  This  dependence 
is  expressed,  for  the  refractive  index,  in  close  approximation  by 
the  formula  (Cauchy's) 


in  which  A  and  B  are  constants  for  one  and  the  same  substance 
while  ^  stands  for  the  wave  length.  Accordingly,  when  for  two 
colors  whose  respective  wave  lengths  are  ^  and  A2  the  refractive 
index  n  has  been  determined,  the  constants  A  and  B  may  be 
calculated  (by  substituting  these  known  values  in  the  above  for- 
mula and  solving  the  two  equations  for  A  and  B),  and  one  then 
knows  the  n  proper  to  any  color  having  another  L  From  this 
dispersion  formula  it  is  seen  that  the  refractive  index,  and  there- 
fore also  the  deviation,  is  the  less,  the  greater  the  wave  length  of 
the  light  refracted.  Since  according  to  page  15  the  red,  of  all  the 
colors  of  the  spectrum,  has  the  greatest  wave  length,  it  is  re- 
fracted the  least;  orange,  yellow,  green,  blue  are  more  strongly 
refracted,  and  the  color  deflected  the  farthest  of  the  whole  spec- 
trum is  violet.  So  this  latter  color  -consists  of  vibrations  of  the 
least  wave  length  that  can  be  perceived  as  light.  If  we  look  at 
such  a  spectrum  (e.g.  one  produced  by  a  prism  of  glass  —  see 
Plate  I,  Fig.  i),  we  see  that  each  of  the  above-named  colors 
still  occupies  a  certain  portion  of  the  length,  still  contains  rays 
therefore  whose  deviation,  and  thus  whose  wave  length,  is  differ- 
ent within  certain  limits.  This  being  the  case,  an  accurate  de- 
termination of  the  refractive  index  for  any  color,  as  yellow  in 
general,  can  not  be  made,  but,  of  all  the  rays  belonging  to 


REFRACTION   OF   LIGHT  47 

the  yellow,  only  for  those  of  one  definite  wave  length,  which  ac- 
cordingly appear  only  in  one  definite  position  in  the  yellow  of 
the  spectrum.  Such  light,  which  consists  only  of  rays  of  one 
and  the  same  wave  length,  is  spoken  of  as  homogeneous  or  as 
monochromatic  light.  We  can  obtain  such  light  in  various 
ways:  — 

1.  The  light  emitted  by  the  incandescent  vapor  of  certain 
metals  is  monochromatic.    For  example,  the  sulphates  of  lithium, 
sodium,  or  thallium,  fused  on  a  thin  platinum  wire  and  vaporized 
in  the  non-luminous  flame  of  a  Bunsen  burner,  emit  monochro- 
matic light:  in  the  case  of  lithium,  red  light  with  a  wave  length 
(in  air)  of  0.000670  mm.;    in  that  of  sodium,  yellow  light  (X  = 
0.000589  mm.);  in  that  of  thallium,  green  (A"*=  0.000535  mm.). 
The  light  of   such  colored  flames  is  the   most  convenient   aid 
to  determining  the  refractive  index  of  a  substance  for  different 
colors. 

2.  If  one  causes  electricity  to  be  discharged  in  a  so-called 
Geissler's  tube  filled  with  hydrogen,  one  obtains  light  which  is  a 
mixture  of  only  three  homogeneous  colors  (^  =  0.000656,  0.000486, 
0.000434  mm.);   and  on  prismatic  decomposition  such  light  ac- 
cordingly exhibits  no  continuous  color  spectrum,  but  only  three 
bright  lines,  one  red  and  two  blue,  which  may  be  used  for  accu- 
rate adjustments.     This  so-called  "  line  spectrum  "  of  hydro- 
gen may  therefore  be  employed  with  advantage,  in  determin- 
ing the  refractive  index  of  a  prism,  when  the  electric  current 
from  an  induction-coil  is  available. 

3.  Sunlight   does  not   contain   all   the   colors   between   the 
extreme  red  and  the  extreme  violet  as  they  are  exhibited,  for  ex- 
ample, by  the  light  of  a  white  flame  when  decomposed  by  a  prism, 
but  in  their  passage  through  the  sun's  outer  atmosphere  certain 
kinds  of  light  are  annihilated.     If  the  bright  spectrum  is  suitably 
produced  there  accordingly  appear,  in  the  positions  correspond- 
ing to  these  kinds  of,  light,  exceedingly  narrow  dark  breaks  (the 
so-called  Fraunhofer's  lines),  whereby  certain    positions  in  the 
spectrum  are  sharply  defined;  and  for  these  positions  the  devia- 


48  OPTICALLY   ISOTROPIC   BODIES 

tion,  and  thus  the  refractive  index  of  the  prism,  can  then  be 
very  accurately  measured.  The  most  important  of  these  lines 
are  inserted  in  Fig.  i,  Plate  I,  with  their  customary  designation 
by  letters  and  specification  of  their  wave  length  (in  millionths  of 
a  millimeter  —  ///z) .  Several  of  them  lie  in  the  positions  occu- 
pied by  the  bright  lines  of  the  emission  spectrum  of  incandes- 
cent gases  and  vapors.  (Sodium  corresponds  to  the  line  D\ 
hydrogen  to  C,  F,  G.) 

4.  Light  is  approximately  monochromatic  when  it  has 
passed  through  certain  colored  bodies;  e.g.  through  red  glass, 
which  annihilates  all  kinds  of  light  except  the  red.  Yet  the 
transmitted  rays  still  have  perceptibly  different  wave  lengths,  so 
that  in  the  spectrum  they  appear  not  in  one  single  position  but 
within  a  transverse  strip  of  some  breadth,  and  therefore  are  not 
adapted  for  accurate  measurements  of  the  deviation. 

By  determining  the  refractive  index  of  transparent  solids  at 
their  boundary  with  air  it  is  found,  as  already  mentioned,  that 
the  light  velocity  in  them  is  in  general  less  than  in  air;  that 
accordingly  the  refraction  quotient  for  passage  from  air  into  such 
a  body  is  greater  than  unity.  With  the  large  majority  it  lies 
between  1.4  and  1.7,  but  there  are  also  substances  whose  refrac- 
tive index  is  considerably  higher.* 

*  For  example,  the  refractive  index  of  the  diamond  is  — 
n  =  2.4135  for  red, 
=  2.4195  for  yellow, 
=  2.4278  for  green. 

So  for  this  body  the  refractive  index  for  light  emerging  from  it  into  air  is  about 
2*1,  or  approximately  the  sine  of  25°.  Therefore  a  light  ray  transmitted  within  a 
diamond  and  striking  a  boundary-face  at  an  angle  of  incidence  greater  than  25° 
can  no  longer  have  a  refracted  ray;  such  a  ray  is  accordingly  totally  reflected,  and, 
because  of -the  high  refractive  index,  the  critical  angle  of  total  reflection  is  so  small 
that  within  a  diamond  bounded  by  many  plane  facets  a  ray  of  light  has  usually  to 
suffer  many  total  reflections.  This  is  the  cause  of  the  numerous  reflections  of 
light  within  a  cut  diamond. 


POLARIZATION  BY   REFLECTION   AND   REFRACTION         49 
POLARIZATION  OF  LIGHT  BY  REFLECTION  AND  REFRACTION 

In  the  reflection  of  light  a  change  in  its  constitution  takes 
place  in  so  far  as,  of  a  ray  of  ordinary  light,  —  which  according 
to  page  16  contains  vibrations  in  all  possible  azimuths,  —  it  is 
chiefly  the  vibrations  perpendicular  to  the  plane  of  incidence 
that  are  thrown  back.  This  occurs  especially  at  a  certain  angle 
of  incidence,  wherefore  by  reflection  at  this  angle,  the  so-called 
"  polarizing  angle  ",  one  obtains  light  the  greater  part  of  which 
is  plane-polarized.  The  light  can  be  polarized  more  completely 
if  we  cause  it  to  be  reflected  from  a  number  of  thin  glass  plates 
laid  one  upon  the  other,  because  then  at  each  single  plate  there 
takes  place  reflection  of  the  light  that  has  penetrated  through 
those  lying  above  it,  with  polarization  of  the  same.  Such  a 
bundle  of  thin  glass  plates  therefore  presents  a  very  convenient 
means  of  producing  from  ordinary  light,  by  reflection  at  the 
polarizing  angle,  light  that  is  plane-polarized.  The  plane  of 
polarization  (see  p.  15)  of  the  reflected  light  is  parallel  to  the  plane 
of  incidence. 

In  refraction,  likewise,  the  ordinary  light  suffers  polarization 
in  a  slight  degree;  here,  however,  it  is  not  the  rays  whose  vibra- 
tions take  place  perpendicular  to  the  plane  of  incidence,  but 
those  vibrating  in  it,  that  are  specially  favored. 

In  the  special  case  of  total  reflection  the  two  phenomena  are 
united,  and  therefore  polarization  of  the  ordinary  light  does  not 
occur.  A  plane-polarized  light  ray,  on  the  other  hand,  is  re- 
solved into  two  mutually  perpendicular  vibrations,  of  which 
one  lies  in  the  plane  of  incidence  and  the  other  in  the  plane  per- 
pendicular to  this  plane;  at  the  same  time  the  two  vibrations 
acquire  a  difference  of  path,  which  depends  on  the  nature  of  the 
medium  and  on  the  angle  at  which  the  rays  fall  upon  the  totally 
reflecting  surface;  the  result  is  that  after  the  total  reflection 
the  previously  plane-polarized  light  in  general  exhibits  elliptical 
polarization.  If  the  vibration  plane  of  the  incident  light  ray 
forms  45°  with  the  plane  of  incidence,  the  two  vibrations  aris- 


DOUBLE    REFRACTION    OF   LIGHT 


ing  on  the  resolution  have  equal  amplitude;  and  if,  of  those  two 
vibrations,  the  path  difference  arising  on  the  total  reflection  is 
then  \X,  their  combination  must  according  to  page  20  result  in 
a  circular  vibration.  On  this  principle  depends  the  use  of  the 
so-called  Fresnel's  rhomb,  abed  (Fig.  26),  for  the  production  of 
circularly  and  of  elliptically  polarized  light. 
The  rhomb  (really  a  parallelepiped)  consists 
of  a  kind  of  glass  for  which  the  difference  of 
path  with  total  reflection  at  a  certain  angle 
corresponds  to  J-^.  Therefore,  if  a  light 
ray  PP'  is  caused  so  to  fall,  as  represented 
in  the  figure,  that  a  total  reflection  at  that 
angle  twice  takes  place,  the  two  arising  vibra- 
tions acquire  a  path  difference  of  \L  When 
the  incident  light  is  plane-polarized,  then  if 
its  vibration  direction  includes  45°  with  the  plane  of  incidence  a 
circularly  polarized  ray  emerges  from  the  Fresnel's  prism;  but 
if  the  vibration  plane  of  the  entering  light  has  a  different  direc- 
tion, the  two  arising  vibrations  are  unequal  in  amplitude  and 
yield  an  elliptically  polarized  ray. 


Fig.  26. 


DOUBLE   REFRACTION  OF  LIGHT 

A  ray  of  ordinary  or  of  plane-polarized  light  falling  perpen- 
dicularly on  the  surface  of  an  optically  isotropic  body,  a  body 
such  as  those  forming  the  subject  of  the  considerations  up  to 
this  point,  suffers  no  refraction  (i  =  o;  so  r  =  o),  nor  does  it  if 
it  leaves  the  body  through  a  face  that  is  parallel  to  the  first  face. 
Now  if  we  investigate  the  character  of  the  ray  after  its  exit,  we 
find  in  the  first  case  that  it  consists,  as  before,  of  ordinary  light; 
in  the  second,  that  it  has  remained  plane-polarized  and  has 
suffered  no  change  in  its  vibration  direction.  This  is  the  natural 
consequence  of  the  isotropy  of  the  luminiferous  ether  in  such  a 
body,  as  is  really  understood  from  the  following.  Let  o  (Fig. 
27)  be  the  point  at  which  a  ray  of  ordinary  light  falls  perpen- 


DOUBLE    REFRACTION    OF   LIGHT 


Fig.  27. 


dicularly  on  the  surface  of  incidence  (represented  by  the  plane 
of  the  figure)  of  the  optically  isotropic  body.     If  at  a  certain 
instant  an  ether  particle  at  o  is  displaced  in  vibrations  along  the 
direction  aa' ,  then  with  every  suc- 
ceeding   vibration    the    vibration 
direction  will  be   rotated   a  very 
small    angle;    for   example,   at    a 
certain  instant  it  will  lie  parallel 

to   cc' .     An   impulse  moving  the  ^ 

ether  particle  o  to  the  point  c 
has  just  the  same  effect  as  two 
impulses,  exerted  simultaneously 
toward  a  and  b,  that  by  themselves 
would  have  moved  the  particle  to 
a  and  /?  respectively.  But  in  the 
medium  in  question  vibrations  are 
transmitted  equally  fast,  whatever  the  direction  in  which  they  take 
place;  consequently, at  every  particle  following  beyond  o  in  the  line 
of  transmission,  that  is  caught  up  by  the  motion,  the  two  impulses 
toward  a  and  b  will  arrive  simultaneously  and  so  will  combine  to 
form  a  single  motion  to  c.  The  same  applies  to  every  vibra- 
tory motion;  and  accordingly,  a  motion  that  takes  place  in  all  azi- 
muths in  rapid  succession  must,  after  penetrating  a  medium  that 
fulfills  the  above-stated  conditions,  keep  up  the  same  change  of 
its  vibration  direction.  Thus  is  it  explained  that  in  such  a 
body  a  ray  of  ordinary  light  is  transmitted  as  ordinary  light; 
that  a  plane-polarized  light  ray  of  any  vibration  direction 
whatsoever  will  in  such  a  medium  be  transmitted  with  un- 
changed vibration  direction,  follows  as  a  matter  of  course. 

A  light  ray  must  behave  differently,  however,  when  it  enters 
an  optically  anisotropic  body  — a  body  whose  ether  is  an  aniso- 
tropic  medium  and  in  which  therefore  differently  directed 
vibrations  are  transmitted  with  different  velocity.  Now  it  is 
learned  empirically  that,  in  all  homogeneous  media  in  which 
the  ether  behaves  differently  in  different  directions,  among  all 


52  DOUBLE   REFRACTION   OF   LIGHT 

the  vibrations  that  take  place  in  any  one  plane  those  having  a 
certain  direction  are  transmitted  the  most  rapidly,  while  those 
vibrations  whose  direction  is  perpendicular  to  this  direction  are 
transmitted  the  most  slowly.     Hence,  if  o  again  be  an  ether  par- 
ticle struck  by  a  motion  having  the  nature  of  ordinary  light, 
just  entering  the  body,  and  if  at  a  certain  instant  the  vibration 
direction  be  the  one,  aa',  that  corresponds  to  the  greatest  trans- 
mission velocity  and  at  some  subsequent  instant  cc',  then  accord- 
ing to  the  above  the  latter  vibration  must  be  equivalent  to  the 
two  vibrations  effected  by  two   simultaneous  impulses;    these 
two  vibrations  take  place  the  one  parallel  to   aa',   the  other 
parallel  to  W   (the  vibration  direction  of  the  rays  transmitted 
the  most  slowly  of  all  those  vibrating  in  the  plane  ab),   and 
their  amplitudes  stand  in    the   ratio  of  the   lengths  oa  and  oft, 
But  these  two  vibrations  are  transmitted  with  different  velocity, 
and  therefore  at  one  of  the  ether  particles  later  set  in  motion 
they  arrive  at  different  times;    there,  accordingly,  they  can  no 
longer  combine  to  form  one  resultant  vibration  cc',  but  will  con- 
tinue their  way  separately.     At  a  still  later  instant  the  particle 
o  will  be  moved  in  a  direction  forming  a  still  greater  angle  with 
aa'  than  does  the  direction  cc' ;    so  this  motion  likewise  will  be 
resolved  into  two,  parallel  to  aa'  and  bbf  respectively;    but   in 
this  case  the  component  parallel  to  W  has  a  greater  value  than 
before.     Thus  the   entire  motion  of  a  ray  of  ordinary   light, 
whose  vibration  plane  very  rapidly  changes,  will  be  resolved  into 
two  light  rays  whose  intensity  varies  with  the  azimuth  of  the 
entering  light.     One  of  these  rays  vibrates  parallel  to  aa',  and 
at  the  instant  when  the  entering  light  vibrates  parallel  to  this 
direction  the  intensity  of  t.his  ray  equals  the  full  intensity  of 
that  light;    the  intensity  then  diminishes,  becomes  zero  as  soon 
as  the  entering  vibration  takes  place  parallel  to  bb',  then   in- 
creases again,  and  so  on.     But  this  variation  of  intensity  occurs 
so  rapidly  that  the  light  ray  produces  the  impression  of  a  ray 
of  constant,  intermediate  intensity,  — of  a  ray  therefore  whose 
intensity  is  half  that  of  the  entering  ray.     As  for  the  second  ray 


DOUBLE   REFRACTION    OF  LIGHT  53 

arising  by  the  resolution,  whose  vibrations  take  place  parallel 
to  W,  this  ray,  at  the  same  instants,  has  the  intensity  zero  when 
that  of  the  first  is  maximum,  increases  in  intensity  as  the  first 
diminishes,  reaches  the  same  maximum  as  did  the  first  ray, 
when  that  ray  has  zero  intensity,  and  so  on;  that  is,  this  second 
ray  likewise  produces  the  impression  of  a  ray  having  the  constant 
brightness  equal  to  half  that  of  the  entering  ray. 

When,  therefore,  a  ray  of  ordinary  light  enters  such  ether, 
anisotropic  ether,  there  arise  from  it  two  light  rays  of  unequal 
transmission  velocity  but  equal  brightness,  whose  vibrations  take 
place  in  the  plane  standing  perpendicular  to  the  ray;  those  of  the 
one  ray  take  place  in  the  direction  corresponding  to  the  greatest 
light  velocity  of  the  rays  vibrating  in  that  plane,  those  of  the 
other  in  the  direction  that  corresponds  to  the  least  of  these  light 
velocities.  Thus  we  obtain  two  plane-polarized  rays  polarized 
perpendicularly  to  each  other,  one  of  which  suffers  a  certain  re- 
tardation as  compared  with  the  other  so  that  the  two  rays,  when 
they  emerge,  have  a  certain  difference  of  path,  according  to  the 
difference  between  their  transmission  velocities  in  the  optically 
anisotropic  body  and  to  the  length  of  their  path  in  it.  The  same 
is  the  case  also  if  the  entering  light  was  plane-polarized;  except 
that  then  the  brightness  of  the  two  rays  is  in  general  not  equal, 
but  depends  on  the  angle  formed  by  the  vibration  plane  of  the 
entering  light  with  the  two  directions  parallel  to  which  the 
vibrations  of  those  rays  take  place  that  are  transmitted  with 
the  greatest  and  the  least  velocity. 

If  the  face  through  which  the  light  emerges  from  the  body  is 
not  parallel  to  the  face  of  entrance,  and  the  light  consequently 
refracted,  then,  since  the  amount  of  refraction  depends  on  the 
light  velocity  in  the  body,  the  deviation  of  the  two  rays  will  be 
different;  so  the  light  ray,  previously  single,  will  in  such  a 
medium  be  "  doubly  refracted  "  into  two  rays  of  different  direc- 
tion. Therefore  the  optically  anisotropic  substances  are  desig- 
nated also  as  doubly  refracting  or  as  birefringent,  the  optically 
isotropic  as  singly  refracting. 


54  DOUBLE   REFRACTION   OF   LIGHT 

ACCORDING  TO  THEIR  BEHAVIOR  TOWARD  LIGHT  RAYS  CRYSTALS 
FALL  INTO  TWO  CLASSES,  THE  SINGLY  REFRACTING  AND  THE  DOUBLY 
REFRACTING.  The  former,  like  amorphous  bodies,  have  for  the 
rays  of  a  definite  color  the  same  refractive  index,  in  whatever  direc- 
tion the  rays  pass  through  the  crystal.  Optically,  therefore,  the 
singly  refracting  crystals  behave  wholly  as  do  amorphous  bodies, 
and  to  these  crystals  applies  all  that  has  been  said  up  to  the  pres- 
ent, while  the  doubly  refracting  shall  form  the  subject  of  what 
now  follows. 

To  distinguish  between  singly  refracting  and  doubly  refract- 
ing crystals  observation  of  the  different  deviation  of  the  two 
light  rays  arising  in  the  latter  can,  in  general,  not  be  availed  of, 
because  the  difference  is  often  so  slight  that  with  moderately 
small  crystals  it  escapes  perception.  It  is  possible,  however, 
even  with  crystals  of  the  smallest  dimensions  and  of  very  feeble 
double  refraction,  to  distinguish  the  two  kinds  from  each  other 
by  the  circumstance  that  the  two  polarized  rays  arising  in  a 
crystal  of  the  latter  kind  emerge  from  it  with  a  difference  of 
path,  although  a  very  slight  one.  In  consequence  of  this  the 
two  rays  may  be  caused  to  interfere  (see  p.  16),  provided  it  is 
possible  to  make  them  vibrate  not  in  two  mutually  perpendicu- 
lar planes,  but  in  the  same  plane.  This  is  accomplished  by  in- 
troducing into  the  path  of  the  two  rays  emerging  from  the 
crystal  a  medium  which  permits  the  passage  only  of  rays  having 
one  definite  vibration  direction.  The  same  purpose  may  be 
served  by  reflection  at  the  polarizing  angle  from  a  glass  plate 
or,  better,  from  a  bundle  of  thin  glass  plates.  (See  p.  49.) 
But  it  is  more  effective  to  pass  the  light  through  a  doubly  re- 
fracting crystal  in  which  one  of  the  rays  arising  therein  is  elimi- 
nated, so  that  only  plane-polarized  light  of  a  definite  vibration 
direction  is  allowed  to  pass  through.  This  is  the  case,  for  ex- 
ample, with  certain  varieties  of  the  mineral  tourmaline,  and  in 
a  still  higher  degree  with  the  crystals  of  iodo-quinine  sulphate 
(the  so-called  herapathite) ,  by  which  one  of  the  two  rays  arising 
in  consequence  of  the  double  refraction  is  so  strongly  absorbed 


DOUBLE   REFRACTION   OF   LIGHT  55 

that  for  this  ray  crystals  of  only  a  slight  thickness  are  quite 
opaque.  But,  since  at  the  same  time  the  other  ray  is  considerably 
weakened  and  colored,  a  uniformly  polarized  ray  is  produced  to 
greater  advantage  by  using  a  colorless  crystal  that  has  strong 
double  refraction;  that  is  to  say,  in  which  the  transmission 
velocities  of  the  two  vibrations  taking  place  perpendicularly  to 
each  other  are  so  different  that  with  a  certain  arrangement  one 
vibration  is  totally  reflected  within  the  crystal,  and  only  the  other 
transmitted.  Such  a  contrivance,  which  in  a  later  section  will  be 
described  in  more  detail  (see  p.  99),  was  constructed  first  by 
Nicol,  from  calcite,  and  after  him  it  is  called  a  Nicol  prism  or 
briefly  a  nicol.*  Since  with  its  aid  one  can  transform  incident 
ordinary  light  into  plane-polarized  light  of  a  definite  vibration 
direction,  such  a  contrivance  is  spoken  of  also  as  a  "  polarizer  ". 
(Cf.  p.  58.) 

If,  then,  we  bring  a  nicol  into  the  path  of  the  two  light  rays 
emerging  from  a  doubly  refracting  crystal,  each  ray  is  resolved 
in  the  polarizer  into  two  rays  of  which  only  one,  the  one  falling 
to  the  definite  vibration  plane  of  the  polarizer,  passes  through; 
so  the  two  vibrations,  previously  perpendicular  to  each  other, 
are  brought  back,  as  it  were,  to  one  vibration  plane. 

But  the  production  of  interference  phenomena  by  this  means 
depends  on  still  a  second  condition,  as  will  be  seen  from  the 
following:  - 

In  Fig.  28  (p.  56)  let  PPf  be  the  vibration  direction  of  a  plane- 
polarized  ray  entering  a  doubly  refracting  crystal  in  the  direc- 
tion perpendicular  to  the  plane  of  the  figure  and  there  resolved 
into  two  rays  having  the  vibration  directions  RE!  and  SS' '. 
Supposing  the  crystal  to  be  of  exactly  such  a  thickness  that  the 
two  rays  transmitted  in  it  with  unequal  speed  have  on  their 
exit  a  path  difference  of  one  whole  wave  length,  or  of  a  multiple 
of  one  wave  length,  of  the  monochromatic  (homogeneous)  light 
employed,  then  as  soon  as  the  light  has  left  the  crystal  an  ether 

*  [The  calcite  crystals  employed  for  these  polarizing  prisms  (as  well  as  for 
similar  optical  purposes)  are  of  the  large,  transparent  variety  known  as  Iceland  spar.] 


DOUBLE    REFRACTION    OF   LIGHT 


particle  lying  in  its  path  will  in  virtue  of  one  of  the  two  arising 
vibratory  motions  always  move  from  O  to  r  at  the  same  instant 
when  the  other  of  the  two  arising  vibrations  would  carry  it  from 
O  to  s.  If  by  means  of  a  nicol  transmitting  only  such  vibra- 
tions as  are  parallel  to  PP'  the  two  rays  are  now  brought  back 
to  one  polarization  plane,  then  of  each  motion  only  that  compo- 
nent passes  through  that  is  parallel  to  PP';  namely,  Op  in  the 
one  case,  Ocr  in  the  other.  These  two  partial  motions,  since 
they  take  place  simultaneously  in  the  same  direction,  must 

simply  combine  to  form  the  sum  of 
their  respective  amplitudes;  must 
interfere  therefore  with  like  state  of 
vibration, — that  is,  with  the  same 
path  difference  they  acquired  in  the 
crystal.  When  on  the  other  hand 
the  thickness  of  the  crystal  is  such 
that  the  retardation  one  of  the  two 
rays  suffers  as  compared  with  the 
other  amounts  to  J  X  or  to  an  uneven 
multiple  of  J-A,  the  motion  of  the 
ether  particle  O,  at  the  instant  when 
in  virtue  of  the  first  vibration  it  is 
moved  to  r,  takes  place  in  virtue  of  the  second  to  sf.  The  two 
motions  can  be  made  to  interfere  by  reducing  them  to  the  same 
vibration  plane  with  the  aid  of  a  nicol.  If,  as  before,  this  so 
stands  that  it  transmits  only  the  vibrations  parallel  to  PP',  the 
interfering  motions  take  place  simultaneously  in  opposite  direc- 
tions; namely,  along  Op  and  Oo'.  So  they  interfere  with  opposite 
state  of  vibration;  that  is,  as  before,  with  the  same  path  difference 
with  which  they  emerged  from  the  crystal.  Hence  it  follows  that, 
stated  generally,  TWO  RAYS  ARISING  FROM  ONE  PLANE-POLARIZED 

RAY  BY  DOUBLE  REFRACTION  INTERFERE  WITH  THE  SAME  PATH 
DIFFERENCE  THEY  ACQUIRED  FROM  THE  DOUBLY  REFRACTING 
CRYSTAL,  PROVIDED  THEY  ARE  BROUGHT  BACK  TO  A  VIBRATION 
PLANE  PARALLEL  TO  THAT  OF  THE  FIRST  RAY. 


R' 


DOUBLE    REFRACTION    OF   LIGHT 


57 


Let  us  now  consider  the  opposite  case,  in  which  the  vibra- 
tion plane  of  the  light  ray  entering  the  crystal  stands  perpendic- 
ular to  the  plane  to  which  the  Nicol  prism  reduces  the  two  rays. 
Again  supposing  PPf  (Fig.  29)  parallel  to  the  vibration  plane  of 
the  entering  light,  and  supposing  the  crystal  to  be  of  such  a 
thickness  that  the  two  rays  arising  therein  have  on  emerging  X 
or  a  multiple  of  \  difference  of  path,  then  by  the  one  of  the  rays 
the  ether  particle  O  will  be  moved  to  r,  by  the  other  simultane- 
ously to  s.  But  in  the  case  now  before  us,  where  the  nicol  is 
rotated  90°  from  its  former  position, 
its  vibration  plane  (that  of  the  light 
it  transmits)  thus  becoming  parallel 
to  QQ'}  the  two  interfering  com- 
ponents are  Op  and  Oo'\  these  lie 
in  opposite  directions;  so  the  inter-  Q,. 
ference  takes  place  with  opposite 
state  of  vibration,  although  the  two 
rays  had  emerged  from  the  crystal 
with  like  state  of  vibration.  When 
on  the  other  hand  the  thickness 
of  the  crystal  is  such  that  the  two 
rays  on  emerging  are  mutually 
opposite  in  their  state  of  vibration,  the  simultaneous  magni- 
tudes of  their  motions  (see  Fig.  29)  are  Or  and  Os'\  after  these 
latter  motions  are  reduced  to  the  single  vibration  plane  QQ' 
their  mutually  interfering,  equal  components,  Op  and  Ooy  lie 
in  the  same  direction;  so  the  interference  takes  place  with  like 
state  of  vibration.  Stated-  generally:  Two  RAYS  ARISING  FROM 

ONE  PLANE-POLARIZED  RAY  BY  DOUBLE  REFRACTION  INTERFERE 
WITH  THE  OPPOSITE  DIFFERENCE  OF  PATH  FROM  THAT  WITH 
WHICH  THEY  EMERGED  FROM  THE  DOUBLY  REFRACTING  CRYSTAL, 
IF  THE  VIBRATION  PLANE  TO  WHICH  THEY  ARE  REDUCED  STANDS 
PERPENDICULAR  TO  THE  ORIGINAL  VIBRATION  DIRECTION. 

If  we  now  imagine,  instead  of  a  plane-polarized  ray,  a  ray 
of  ordinary  light  entering  the  crystal,  then  at  the  instant  when 


58  DOUBLE   REFRACTION   OF   LIGHT 

the  vibration  plane  of  this  entering  ray  is  parallel  to  that  of 
the  reducing  nicol  the  interference  takes  place  with  the  same 
path  difference  with  which  the  two  arising  rays  emerge  from  the 
crystal;  while  after  an  exceedingly  short  space  of  time,  during 
which  the  vibration  plane  of  the  ordinary  light  has  rotated  90°, 
the  two  rays  interfere  with  the  opposite  path  difference,  —  i.e. 
with  their  path  difference  at  emergence  +  or  —  %L  Conse- 
quently, in  the  path  of  the  light  the  maxima  and  minima  of 
light  intensity  arising  by  the  interference  must  follow  one  an- 
other just  as  rapidly  as  the  polarization  plane  of  the  ordinary 
light  rotates  90°.  Of  this  change  of  intensity  we  shall  therefore 
perceive  just  as  little  as  if  we  viewed  the  rays  directly,  as  on 
page  52  et  seq.;  i.e.  without  passing  the  rays  through  a  nicol  after 
their  exit.  Interference  phenomena  can,  accordingly,  not  be 
observed  in  this  way,  since,  although  at  a  definite  point  the 
interference  does  produce  momentary  darkness,  it  produces  after 
an  immeasurably  short  time  just  so  much  the  greater  brightness, 
so  that  all  taken  together  there  results  the  impression  of  a  ray 
having  a  constant,  intermediate  intensity. 

From  the  foregoing  the  conditions  necessary  for  the  interfer- 
ence of  plane-polarized  light  are  now  fully  understood;  they 
read:  Two  PLANE-POLARIZED  RAYS  INTERFERE  ONLY  WHEN  THEY 

HAVE  ARISEN  FROM  A  SINGLE  PLANE-POLARIZED  RAY  BY  DOUBLE 
REFRACTION  AND  ARE  REDUCED  TO  THE  SAME  POLARIZATION 
PLANE  BY  MEANS  OF  A  NlCOL  PRISM. 

In  order,  then,  to  investigate  a  crystal  for  its  double  refrac- 
tion by  means  oj  the  interference  resulting  from  the  difference 
in  velocity  of  the  doubly  refracted  rays,  we  must  employ  two 
nicols.  One  of  these  serves  to  transform  the  ordinary  light, 
before  it  enters  the  crystal  to  be  investigated,  into  plane-polar- 
ized light,  wherefore  this  first  nicol  is  called  the  polarizer  in  the 
narrower  sense  of  the  word;  the  other  permits  the  light  emerging 
from  the  crystal  to  be  analyzed,  and  is  therefore  designated  as 
the  analyzer.  [Such  a  combination  of  two  nicols  or  other  polar- 
i-zing  device,  used  for  investigating  a  substance  in  polarized 


DOUBLE    REFRACTION    OF  LIGHT  59 

light,  is  generally  called  a  polariscope,  sometimes  a  "polarization- 
instrument  ".  (See  p.  74  et  seq.)] 

If  we  look  through  two  nicols  placed  one  behind  the  otheT" 
in  parallel  position,  the  light  rays  emerging  from  the  one  will 
pass  through  the  second  with  the  same  vibration  direction,  and 
the  brightness  of  the  light  will  accordingly  experience  no  change. 
But  if  we  rotate  the  second  nicol  about  the  direction  of  the  light 
rays  as  an  axis,  only  that  component  will  be  transmitted  that 
corresponds  to  the  vibration  plane  of  this  nicol;  with  further 
rotation  of  the  nicol  this  component  always  becomes  smaller, 
and  finally,  after  90°  rotation,  equal  to  zero,  because  then  the 
light  entering  the  second  nicol  has  exactly  the  vibration  direc- 
tion that  in  the  nicol  is  totally  reflected.  So  when  two  nicols 
are  "  crossed  "  in  this  way  they  transmit  no  light. 

Inserting  a  singly  refracting  crystal  between  the  two  crossed 
nicols  can  produce  no  change,  since  the  light-vibrations  in  such 
a  body,  in  whatever  direction  they  take  place,  are  always  trans- 
mitted in  the  same  way,  without  resolution  into  components. 
BETWEEN  CROSSED  NICOLS,  THEREFORE,  A  SINGLY  REFRACTING 

CRYSTAL    APPEARS    DARK    IN    EVERY     POSITION.       Even    Crystals    of 

microscopic  dimensions  can  be  subjected  to  this  experiment,  by 
placing  a  polarizer  beneath  the  stage  of  a  microscope  and  mount- 
ing an  analyzer  on  the  eyepiece.  If  one  rotates  the  analyzer  90° 
from  the  position  in  which  it  is  parallel  to  the  polarizer,  the 
analyzer  cuts  off  all  the  polarized  light  entering  the  instrument 
from  below;  the  field  of  the  microscope  becomes  dark,  as  do  all 
singly  refracting  crystals  that  may  be  in  it;  and  moreover,  in 
the  darkness  of  these  crystals  no  change  can  take  place  if  we 
rotate  them  in  their  own  plane  (for  which  purpose  the  prepara- 
tion to  be  investigated  must  be  on  a  rotatable  stage  fitted  to  the 
instrument) . 

Quite  different  is  the  behavior  of  doubly  refracting  crystals. 

If,  as  represented  by  their  cross-sections  in  Fig.  30  (p.  60), 
two  crossed  nicols  one  of  which  transmits  the  vibrations  that  are 
parallel  to  PP',  the  other  those  taking  place  parallel  to  <2Q', 


6o 


DOUBLE   REFRACTION   OF   LIGHT 


are  inserted  one  behind  the  other  (in  the  figure,  for  the  sake  of 
clearness,  they  are  placed  side  by  side)  in  the  path  of  the  light 
rays,  and  if  a  thin  doubly  refracting  crystal,  K,  is  brought  be- 
tween the  two  in  such  a  way  that  the  vibrations,  RR'  and  SS', 
of  the  two  rays  arising  in  it  are  parallel  respectively  to  the 
vibrations  transmitted  by  the  two  nicols,  — then  complete  ex- 
tinction of  the  light  does  indeed  occur,  as  with  a  singly  refract- 
ing crystal;  for  the  vibration  taking  place  parallel  to  PP',  which 
enters  K  from  the  first  nicol,  is  then  exactly  parallel  to  RR', 
wherefore  the  whole  vibration,  since  the  component  falling  to 
the  direction  SS'  is  zero,  passes  through  the  crystal  unaltered 


Fig.  30. 

and  in  the  second  nicol  is  totally  reflected.  This  is  no  longer 
the  case,  however,  so  soon  as  by  rotating  the  crystal  K  in  its  own 
plane  we  bring  it  from  the  above  position  into  another. 

Assuming  the  faces  of  entrance  and  emergence  of  the  light  to 
be  so  close  together,  i.e.  the  crystal  to  have  the  form  of  a  plane- 
parallel  plate  so  thin,  that  on  emerging  from  it  the  two  rays 
of  a  definite  color  polarized  perpendicularly  to  each  other  have 
acquired  a  path  difference  of  exactly  a  half  wave  length,  then, 
when  the  plate  K  has  been  rotated  an  angle  FOR  (Fig.  31), 
there  arise  in  it  two  vibrations  which  on  emerging  take  place 
simultaneously  to  r  and  s'  and  of  which  the  components  that 
pass  through  the  second  nicol,  Op  and  Oa,  are  added  to  each 
other.  Consequently  the  plate  now  appears  bright,  and  its 
brightness  depends  on  the  magnitude  of  the  components  Op 
and  Off.  From  a  comparison  of  Fig's  31,  32,  and  33  it  follows 
that  these  two  components,  which  are  always  of  equal  magni- 


DOUBLE   REFRACTION   OF   LIGHT 


6l 


tude,  attain  their  maximum,  and  that  accordingly  their  sum  is 
greatest,  when  the  rotation  angle  FOR  is  45°.  (Fig.  32.)  After 
a  rotation  of  90°  they  both  become  zero  again;  we  then  have  the 
case  represented  in  Fig.  30,  except  that  the  directions  RR'  and 
SS'  have  exchanged  places.  So  the  crystal  plate  illuminated  by 
light  rays  of  the  color  in  question  becomes  dark  only  when  its 
vibration  directions,  RR'  and  SS',  coincide  with  those,  PPf  and 
QQ',  of  the  nicols;  and  this  is  obviously  the  case  four  times  dur- 
ing one  complete  rotation;  in  every  other  position,  on  the  other 


Fig-  31- 


hand,  it  will  appear  bright,  bright  with  the  color  of  the  light 
used,  and  the  brightness  will  reach  a  maximum  whenever  the 
directions  RR  and  SS'  form  45°  with  PP'  and  QQ'. 

Let  us  now  suppose  the  crystal  to  be  investigated  not  in 
monochromatic,  but  in  white  light.  The  same  retardation  of 
one  of  the  doubly  refracted  rays  as  compared  with  the  other 
will  now,  for  the  vibrations  of  those  colors  that  have  a  greater 
wave  length,  amount  to  less  than  %X  and  for  the  colors  having  a 
lesser  wave  length  to  more  than  ^;  in  both  cases,  therefore, 
the  resultant  from  the  interference  of  the  two  components  paral- 
lel to  PP'  and  QQ'  is  not  equal  to  their  sum,  but  less.  (Cf. 
p.  19.)  Consequently  the  different  colors,  after  they  have 
passed  through  the  second  nicol,  will  no  longer  have  the  same 
relative  intensities  as  previously,  in  the  white  light;  they  will 
then  no  longer  produce  in  our  eye  the  impression  of  white,  but  a 
color  impression  in  which  that  color  predominates  that  on  the 


62  DOUBLE   REFRACTION    OF   LIGHT 

interference  is  most  favored  in  intensity.  Since  this  favoring 
reaches  its  maximum  with  45°  rotation,  it  is  in  this  position 
of  the  plate  that  the  color  in  question  appears  relatively  purest. 
If  the  plate  has  a  different  thickness,  the  same  will  apply  to  a 
different  position  in  the  spectrum,  and  the  plate  will  accordingly 
be  lighted  up  with  a  different  color.  Stated  generally:  - 

A  THIN  DOUBLY  REFRACTING  CRYSTAL  ROTATED  BETWEEN 
CROSSED  NICOLS  APPEARS  DARK  FOUR  TIMES;  IN  THE  INTERMEDI- 
ATE POSITIONS  BRIGHT  WITH  A  DEFINITE  COLOR,  WHICH  IS  PUREST 
AND  BRIGHTEST  IN  THE  FOUR  DIAGONAL  POSITIONS. 

When  the  two  nicols  are  not  crossed,  but  set  parallel,  then 
according  to  page  56  the  interference  takes  place  not  with  the 
opposite,  but  with  the  same  path  difference  as  arose  in  the  crys- 
tal; so  the  rays  for  which  the  retardation  amounts  to  %X  or  an 
uneven  multiple  thereof  do  not  superpose  their  amplitudes,  but 
completely  annihilate  each  other,  while  those  whose  difference  of 
path  amounts  to  one  or  more  whole  wave  lengths  interfere  with 
like  state  of  vibration,  arid  strengthen  each  other.  Thus,  in  the 
case  of  illumination  with  white  light,  between  parallel  nicols  those 
colors  are  destroyed  that  most  predominate  when  the  nicols  are 
crossed,  and  those  colors  predominate  that  with  crossed  nicols 
have  the  least  intensity.  The  composite  color  arising  in  this  way 
is  called  the  complementary  color  of  the  other. 

That  the  colors  exhibited  in  polarized  light  by  thin  plates  of 
doubly  refracting  crystals  are  really  composite  colors  arising  in 
the  manner  described,  may  be  demonstrated  by  decomposing 
them  into  a  spectrum  with  a  prism.  In  the  place  of  the  color 
that  the  interference  destroys,  one  then  sees  a  dark  stripe;  and 
this  fades  out  on  both  sides,  because  after  the  interference  there 
remains,  of  the  adjacent  colors  as  well,  only  a  slight  intensity. 
If  we  rotate  one  of  the  nicols  90°,  the  greatest  brightness  is  now 
seen  where  previously  the  dark  band  appeared,  with  diminishing 
intensity  of  the  light  on  either  side. 

But  the  appearance  of  a  color  when  a  crystal  is  rotated  be- 
tween crossed  nicols  not  only  in  general  distinguishes  that  crys- 


POLARIZATION-COLORS  63 

tal  as  doubly  refracting,  but  the  definite  shade  of  color  observed 
indicates  directly  what  path  difference  the  two  rays  have  ac- 
quired as  compared  with  each  other.*  Now  we  say  of  a  crys- 
tal plate  that  it  has  "  stronger  double  refraction  ",  or  "  higher 
birefringence"  (cf.  p.  101),  than  another,  if  with  the  same  thick- 
ness it  produces  a  greater  difference  of  path  of  the  two  rays 
vibrating  perpendicularly  to  each  other.  Hence  it  is  manifest 
that  with  known  plate  thickness  the  color  shade  observed  per- 
mits a  conclusion  to  be  drawn  as  to  the  strength  of  the  double 
refraction,  or  conversely,  with  known  birefringence  a  conclusion 
as  to  the  thickness  of  the  plate.  On  account  of  this  practical 
importance  in  optical  crystallography  the  colors  corresponding 
to  the  different  retardations  [called  interference-  or  polarization- 
colors],  which  doubly  refracting  crystals  exhibit  in  polarized  light, 
will  in  the  following  section  be  considered  in  detail. 

POLARIZATION-COLORS  OF  DOUBLY  REFRACTING  CRYSTALS 

The  color  shades  appearing  in  polarized  light  as  the  result  of 
interference  are  represented  on  Plate  I,  in  Fig.  2,  where  each  tint 
corresponds  to  that  value  of  J,  the  path  difference  of  the  two 
interfering  rays,  that  is  specified  below  it  in  fifj.  (millionths  of  a 
millimeter). 

*  The  dependence  in  which  the  magnitude  of  this  path  difference  stands  to 
the  thickness  of  the  plate  and  to  the  difference  between  the  refractive  indices  of 
the  two  rays,  is  very  easily  understood.  If  we  denote  the  thicknesses  by  e,  then 
for  the  ray  having  the  wave  length  At  the  number  of  vibrations  in  this  distance  e  is 

y,  and  for  the  ray  having  the  wave  length  A2  it  is  y- .    Therefore  the  difference  in  the 

X|  /2 

number  of  the  two  kinds  of  vibrations  is y  ;  and  the  path  difference  (J)  referred 

1  2 

to  the  wave  length  in  air  (A0)  is 


But,  if  the  refractive  indices  of  the  two  rays  (for  the  crystal  as  compared  with  air) 
are  denoted  by  «x  and  «2,  we  have  nv  =  y,  n2  =  y  ;  and  thence  follows,  on  sub- 
stituting above, 

A  =  e  (w1  —  n2). 


64  DOUBLE    REFRACTION    OF   LIGHT 

The  wave  lengths  of  the  different  colors  of  the  visible  spec- 
trum (see  Fig.  i  on  the  same  plate)  lie  between  the  limits 

0.000380  and  0.000775  mm-> 

since  the  lines  H,  which  lie  inside  the  most  refrangible  violet, 
have  the  wave  lengths  0.000393  and  0.000397  mm.,  while  for 
the  line  A,  lying  inside  the  least  refrangible  red,  A  =  0.000760  mm. 
Hence,  if  we  imagine  a  doubly  refracting  crystal  plate  so 
thin  or  of  a  substance  having  such  low  birefringence  that  the 
two  rays  arising  in  it  acquire  a  path  difference  amounting  to 
only  a  very  small  fraction  of  the  wave  length  of  the  extreme  red, 
this  path  difference  is  likewise,  although  not  so  slight  a  one, 
yet  a  very  small  fraction  of  the  wave  length  of  the  extreme 
violet;  so  the  rays  of  all  colors  emerge  from  the  crystal  vibrating 
in  nearly  the  same  state  of  vibration,  and  between  crossed 
nicols  (cf.  p.  57)  are  almost  completely  annihilated.  Conse- 
quently such  a  crystal  plate  will  exhibit  only  a  very  slight 
brightening,  a  shade  of  "gray,  which  with  increasing  retardation 
naturally  becomes  lighter.  Not  until  the  retardation  amounts  to 
about  o.oooioo  mm.  does  a  pale  violet  tint  (lavender-gray) 
appear.  For  this  quantity  is  about  i^  of  the  extreme  red  and 
about  J^  of  the  extreme  violet;  so  that  for  the  latter  color  the 
difference  of  path  approaches  considerably  closer  to  the  amount 
of  a  half  wave  length,  with  which  value  of  the  same  the  interfer- 
ence between  crossed  nicols  takes  place  with  like  state  of  vibra- 
tion, giving  maximum  brightness.  Relatively,  therefore,  the 
violet  must  have  somewhat  more  brightness  than  the  remaining 
colors  of  the  white;  but,  since  its  absolute  intensity  in  the  spec- 
trum is  very  slight,  the  aggregate  color  impression  produced 
differs  so  little  from  a  white  of  slight  intensity,  i.e.  from  pure 
light  gray,  that  in  the  figure  the  difference  can  hardly  be  repro- 
duced. With  increasing  amount  of  the  retardation  the  same 
applies  to  colors  having  greater  wave  length.  Farther  on 
therefore  blue,  then  green,  then  yellow,  predominate  somewhat 
more  in  the  color  shade  arising  by  the  mixing  but,  since  all  the 


POLARIZATION-COLORS  65 

colors  now  naturally  acquire  greater  intensity  and  since  the  less 
refrangible  are  in  general  the  brighter,  at  first  only  in  a  slight 
degree;  so  that  the  composite  color,  continuously  becoming 
brighter,  still  differs  but  little  from  white,  *  until  finally,  with  a 
path  difference  of  about  0.000300  mm.,  a  vivid  yellow  becomes 
visible.  This  quantity,  as  may  be  learned  from  the  solar  spec- 
trum, Fig.  i,  is  equal  to  J  A  of  a  very  intense  yellow,  and  so 
between  crossed  nicols  this  yellow  interferes  with  like  state  of 
vibration;  the  extreme  violet,  since  the  quantity  mentioned 
amounts  to  almost  one  whole  wave  length  of  this  color,  is  nearly 
annihilated,  while  green  and  red  suffer  by  the  interference 
about  an  equal  weakening;  but  the  mixing  of  these  latter, 
so-called  complementary  colors  gives  white,  so  that  by  them  the 
yellow  coloring  of  the  aggregate  impression  is  influenced  only 
in  so  far  as  a  lighter  yellow  results.  With  increasing  thickness  or 
rising  birefringence  of  the  crystal  plate  the  yellow  passes  over 
into  orange  and  red;  and  the  red  is  most  vivid  when  the  re- 
tardation of  the  one  ray  as  compared  with  the  other  amounts  to 
about  0.000530  mm.  For,  this  quantity  being  the  wave  length 
of  green  near  the  line  E,  between  crossed  nicols  the  green  is 
with  this  retardation  totally  annihilated,  while  the  red  rays  inter- 
fere with  about  f  A  and  the  violet  with  almost  f  ^,  both  the  red 
and  the  violet  therefore  with  nearly  that  path  difference  that 
yields  the  sum  of  the  intensities  of  the  two  combining  rays; 
so  that,  since  into  the  composition  of  white  light  the  red  rays 
enter  with  a  much  greater  intensity  than  do  the  violet,  the  aggre- 
gate impression  is  an  intense  red,  whose  prismatic  decompo- 
sition yields  a  spectrum  having  a  dark  stripe  in  the  green.f 

*  In  passing  over  from  bluish  to  yellowish  white  it  appears  even  pure  white, 
owing  to  the  intensities  of  the  component  colors. 

f  With  parallel  nicols  a  vivid  green  appears,  as  complementary  color,  whose 
spectrum  naturally  exhibits  a  light  maximum  in  the  green  (A  =  530),  the  bright- 
ness diminishing  on  both  sides.  With  the  nicols  in  this  position,  however,  no 
dark  stripe  is  visible,  because  complete  annihilation  would  take  place  only  for 
X  =  265  (=  |  X  530)  and  for  A  =  795  (=  f  X  530),  and  such  wave  lengths  as 
these  fall  outside  the  visible  spectrum. 


66  DOUBLE   REFRACTION    OF   LIGHT 

Farther  on,  this  red  passes  over  into  violet,  the  violet  being  most 
vivid  when  the  retardation  amounts  to  0.000575  mm->  because 
this  is  nearly  equal  to  |>*  of  the  brightest  violet  in  the  spectrum 
and  because  it  is  precisely  then  that  the  most  intense  yellow  (near 
the  line  D)  is  annihilated.  The  interference-color  correspond- 
ing to  this  retardation  is  termed  sensitive  violet,  because  a 
small  variation  of  the  path  difference  in  either  sense  produces  a 
comparatively  great  change  in  the  color,  toward  red  or  blue. 
Since  from  now  on  the  same  colors  in  a  certain  measure  repeat 
themselves,  the  colors  hitherto  considered  are  designated  as 
colors  of  the  first  order;  these  accordingly  are  gray,  lavender- 
gray,  bluish  gray,  greenish  white,  yellowish  white,  yellow  of  the 
first  order,  orange  of  the  first  order,  red  of  the  first  order,  and 
sensitive  violet  of  the  first  order. 

As  may  be  seen  from  the  colored  plate,  the  shades  change,  in 
proportion  to  the  retardation,  far  the  most  rapidly  in  the  entire 
last  part  of  the  colors  of  the  first  order;  therefore,  if  a  thin  doubly 
refracting  crystal  plate  giving  the  red  or  violet  of  the  first  order 
is  inserted  in  the  path  of  the  light  rays  passing  through  a  micro- 
scope provided  with  two  crossed  nicols,  any  crystals  of  very  low 
birefringence  that  may  be  present  in  the  field  of  the  microscope 
will  appear  distinctly  in  a  different  color  from  the  rest  of  the 
field,  because  the  addition  or  subtraction  of  a  small  path  differ- 
ence effected  by  them  at  once  produces  a  perceptible  change  in 
the  tint.  On  this  principle  is  based  the  employment  of  thin 
crystal  plates  of  the  kind  mentioned  (e.g.  of  gypsum),  for  recog- 
nizing exceedingly  feeble  double  refraction.  (Particulars  in  a 
later  section,  on  the  Determination  of  the  Character  of  the 
Double  Refraction.) 

With  still  increasing  thickness  of  the  considered  crystal  plate 
the  violet  passes  gradually  over  into  blue,  which  is  purest  with  a 
path  difference  of  0.000660  mm.;  for  this  value  is  equal  to  f-A  of 
the  brightest  blue  in  the  spectrum  but  equal  also  to  ^  of  the 
brightest  orange,  so  that  after  the  interference  the  blue  is  at 
a  maximum,  while  orange  and  yellow,  which  would  cause  the 


POLARIZATION-COLORS  6/ 

color  mixture  to  appear  green,  are  at  a  minimum.  Not  until 
the  retardation  amounts  to  about  0.000800  mm.  does  a  vivid 
green  appear;  for  this  value  corresponds  to  J  A  of  the  brightest 
green  (at  E),  as  well  as  nearly  to  one  whole  wave  length  of  the 
extreme  red  and  to  twice  that  of  the  violet,  so  that  by  the  inter- 
ference the  colors  lying  at  the  two  ends  of  the  spectrum  are 
totally  annihilated.  Between  850  and  900  /*/*  the  maximum  of 
intensity  moves  to  yellow  and  orange,  wherefore  these  colors 
predominate;  and  farther  on  the  latter  passes  over  more  and 
more  into  red;  until  finally,  with  a  retardation  of  0.001060  mm. 
(i.e.  2  X  0.000530  mm.,  the  retardation  giving  the  red  of  the 
first  order),  the  so-called  red  of  the  second  order  appears  in  its 
purest  shade.  For  the  value  given  is  equal  to  2  A  for  green  near 
the  line  E,  this  color  being  therefore  completely  annihilated,  and 
equal  also  to  f  A  for  indigo  of  the  wave  length  424  and  to  f  A 
for  red  with  the  wave  length  700;  so  that,  since  the  latter  color 
is  considerably  more  luminous  than  indigo,  a  comparatively 
pure  red  appears.  Decomposed  into  a  spectrum,  this  color,  like 
the  red  of  the  first  order,  yields,  as  is  manifest  from  the  fore- 
going, only  a  dark  stripe  in  the  green;  with  parallel  nicols,  on 
the  other  hand,  — and  the  maximum  brightness  then  lies  in  the 
green,  —  two  dark  stripes  appear,  one  in  the  indigo  and  the  other 
in  the  red,  so  that  both  ends  of  the  spectrum  appear  darkened, 
something  which  is  not  the  case  with  the  red  of  the  first  order. 
(See  footnote  f?  p.  65.) 

As  may  readily  be  understood,  with  further  increase  of  the 
retardation  the  red  passes  over  into  violet,  and  this,  from  the 
path  difference  0.001130  mm.  on,  into  blue  (the  transition  color 
in  question  is  called  "sensitive  violet  No.  II");  so  that  here 
there  begins  a  new  series  of  the  colors  blue,  green,  yellow,  red, 
and  violet,  which  are  known  as  colors  of  the  third  order.  The 
red  of  this  order  corresponds  to  a  retardation  of  0.001590  mm.; 
i.e.  again  to  a  whole  multiple  of  X  of  the  green  at  £,  namely,  to 
3  X  0.000530;  accordingly,  with  this  double  refraction  also,  the 
color  named  is  annihilated  by  the  interference.  But  so  too  is  a 


68  DOUBLE  REFRACTION   OF  LIGHT 

part  of  the  red  and  of  the  violet;  for  the  difference  of  path  is 
now,  also,  nearly  equal  to  2^  of  the  extreme  red  and  equal  to 
4^  of  violet.  Therefore  the  colors  having  their  maximum  inten- 
sity after  the  interference  are  those  with  f  A  and  %\  difference 
of  path,  namely,  orange-red  (X  =  636)  and  indigo-blue  (X  =  454) ; 
these  mix  to  form  a  red  that  is  much  less  pure  and  also  less 
intense,  because  at  the  same  time  very  considerable  portions  of 
the  spectrum  are  obliterated.  (The  spectrum  exhibits  a  dark 
stripe  in  the  green,  a  second  in  the  violet,  and  a  shortening  of 
the  red  end;  when  the  nicols  are  set  parallel,  two  dark  stripes 
naturally  appear  in  the  orange-red  and  the  indigo-blue  respec- 
tively.) 

In  the  fourth  order,  now  following,  the  purity  and  the  vivid- 
ness of  the  colors  diminish  in  a  much  higher  degree;  for  the  red 
of  this  order,  for  example,  corresponds  to  a  path  difference  of 
0.002 1 20  mm.,  and  this,  while  equal  to  4  X  530,  is  equal  also  to 
3  X  707  and  to  5  X  424  /*//,  the  former  of  these  values  being  the 
wave  length  of  a  very  vivid  red  and  the  latter  that  of  indigo  near 
the  line  G.  The  spectrum  of  the  red  of  this  order  accordingly 
yields,  besides  the  dark  stripe  in  the  green,  yet  two  more,  one  in 
the  red  and  the  other  in  the  indigo;  so  that  the  mixing  results 
only  in  a  pale  red  strongly  tinged  with  yellow.  In  just  the  same 
way,  the  complementary  color,  arising  with  parallel  nicols,  is  a 
very  pale  and  impure  green,  whose  prismatic  decomposition  ex- 
hibits maxima  at  the  same  three  positions  in  the  spectrum  where 
crossed  nicols  gave  a  dark  stripe,  while  a  dark  stripe  now  appears 
in  both  the  blue  and  the  yellow. 

The  colors  of  the  first,  second,  third,  and  fourth  orders 
represented  in  Fig.  2  of  Plate  I  can  be  best  studied  with  the 
aid  of  a  thin  wedge  of  the  form  shown  in  Fig.  101,  page  202, 
cut  from  a  doubly  refracting  crystal,  as  quartz*;  one  pushes  the 

*  Quartz  wedges  of  excellent  workmanship,  which  exhibit  the  colors  of  the 
first  four  orders,  are  supplied  by  the  firm  of  Steeg  and  Reuter  in  Homburg.  (See 
Appendix.)  But,  in  such  wedges  the  colors  of  the  first  order  begin  with  the  light 
gray  corresponding  to  a  path  difference  of  about  too  ///*,  because  it  is  not  possible 
to  grind  the  quartz  thinner. 


POLARIZATION-COLORS  69 

wedge  between  two  crossed  nicols  in  such  wise  that  its  long  edge 
forms  an  angle  of  45°  with  their  vibration  directions.  But  it 
must  here  be  remarked  that  the  foregoing  explanation  of  the 
interference-colors  of  thin  doubly  refracting  lamellae  presup- 
poses that  the  difference  of  path  produced  by  such  lamellae  be 
the  same  for  all  colors  contained  in  the  white  light.  This  pre- 
supposition, as  will  be  shown  farther  on,  does  in  the  case  of  no 
doubly  refracting  body  prove  to  be  exactly  correct,  the  retarda- 
tion for  the  rays  of  a  different  vibration  period  being  a  little  greater 
or  less.  Variations  are  thereby  produced  in  the  resulting  com- 
posite color,  —  variations  so  slight,  however,  in  the  majority  of 
cases,  that  in  a  reproduction  such  as  is  given  in  Fig.  2  of  Plate  I 
they  can  hardly  be  represented.  Since  the  strength  of  the 
double  refraction,  and  accordingly  also  the  difference  of  path 
arising  with  a  definite  thickness  of  plate,  does  with  certain  sub- 
stances increase  with  increasing  wave  length  of  the  light,  but 
with  others  diminish,  this  figure  may  be  said  to  represent  the 
normal  case;  which  is  the  more  closely  approximated  by  the 
colors  of  the  first,  second,  third,  and  fourth  orders  actually 
observed  in  crystals,  the  less  the  birefringence  of  the  crystals 
differs  for  different  colors. 

The  greater  the  thickness  becomes,  of  the  crystal  plate 
viewed  in  the  polarized  light,  or  the  higher  its  birefringence,  — 
the  greater  therefore  the  retardation  of  the  two  rays  arising  in 
it,  as  compared  with  each  other,  —  the  less  may  the  colors  of 
the  fifth,  sixth,  and  higher  orders  resulting  from  the  interference 
differ  from  white,  as  is  taught  by  a  simple  consideration.  Sup- 
pose, for  example,  the  difference  of  path  to  be  0.011475  mm- 
This  quantity  is  equal  to  15  X  765  ////,  i.e.  to  15  X  of  the  extreme 
red  beyond  the  line  A',  but  likewise  equal  to  the  following: 

16  X  717  ftp,  or  16  X  of  the  red  at  the  line  a 

17  X  675  fifjL,  or  17  ^  of  the  red  between  B  and  C 

18  X  637  fifty  or  18  A  of  orange 

19  X  604  jj.fi,  or  19  A  of  the  yellow  between  line  D  and  orange 

20  X  574  fift,  or  20  X  of  the  yellow  on  the  opposite  side  of  D 


70  DOUBLE    REFRACTION    OF    LIGHT 

21  x  547  /*//,  or  21  X  of  yellowish  green 

22  X  522  /Aft,  or  22  X  of  the  green  between  b  and  E 

23  X  499  up,  or  23  ^  of  bluish  green 

24  X  478  /if*,  or  24  X  of  the  blue  near  F 

25  X  459  ftp,  or  25  A  of  the  most  vivid  blue 

26  X  441  {JL/JL,  or  26  ^  of  the  blue  near  the  line  G 

27  X  425  /*/*,  or  27  A  of  indigo 

28  X  410  ////,  or  28  A  of  the  transition  to  violet 

29  X  395  /*/*,  or  29  A  of  the  violet  at  the  line  H 

If  a  crystal  plate  of  this  kind  is  illuminated  with  white  light, 
and  if  after  the  interference  the  emerging  rays  are  decomposed 
by  prism,  then  in  their  spectrum  there  accordingly  appear  fifteen 
dark  stripes,  while,  of  the  colors  lying  between,  those  for  which 
the  path  difference  amounts  to  an  uneven  multiple  of  a  half 
wave  length  have  their  maximum  brightness.  Therefore  the 
arising  mixture  contains  all  colors  of  the  spectrum,  containing 
them,  too,  in  nearly  the  same  relative  intensities  as  did  pre- 
viously the  white  light.  The  consequence  is  that  in  our  eye  this 
mixture  also  will  produce  the  impression  of  white,  —  or,  to  be 
more  correct,  of  light  gray,  since  by  the  interference  extinction 
has  taken  place  for  portions  of  all  the  colors,  the  total  intensity 
thus  naturally  becoming  less  considerable.  This  interference- 
color  is  called  white  of  a  higher  order.  If  we  rotate  the  nicols 
into  parallel  position,  then  in  the  spectrum  dark  stripes  appear 
at  all  points  where  with  crossed  nicols  there  were  maxima  of 
light,  and  bright  ones  where  previously  the  light  was  minimum; 
but  the  arising  interference-color,  the  complementary  color  of 
the  last,  can,  of  course,  without  spectral  decomposition,  just  as 
little  as  that  last  color  be  distinguished  from  white.  The  thicker 
a  plate  that  exhibits  the  white  of  a  higher  order,  the  more 
numerous  are  the  positions  in  the  spectrum  for  which,  by  inter- 
ference, the  annihilation  takes  place,  wherefore  the  more  ex- 
actly do  the  remaining  colors  possess  the  same  relative  intensities 
as  in  the  white  light  and  the  more  exactly  are  their  intensities 
equal  to  one-half  what  they  would  be  without  the  interference. 


POLARIZATION-COLORS  7 1 

Hence  it  follows  that,  from  a  certain  thickness  on,  a  doubly 
refracting  crystal  plate  rotated  between  crossed  nicols  no  longer 
exhibits  a  color  but  simply  becomes  light  and  dark;  the  greatest 
brightening  with  white  light  takes  place  when  the  vibration 
directions  of  the  plate  are  in  diagonal  position  to  those  of  the 
nicols.  The  thickness  with  which  the  white  of  a  higher  order 
appears  depends  on  the  birefringence  of  the  crystallized  sub- 
stance in  question,  i.e.  on  the  specific  strength  of  its  double  re- 
fraction; for,  obviously,  the  greater  this  is,  with  so  much  the 
less  thickness  of  a  plate  composed  of  the  substance  is  the  re- 
quisite difference  of  path  attained. 

As  may  be  understood  from  the  foregoing  directly,  the  color 
observed  in  thin  crystal  plates  suffers  a  perceptible  change  when 
there  is  a  small  variation  in  the  thickness;  consequently  such  a 
plate,  if  at  different  points  it  has  not  exactly  the  same  thickness, 
does  not  exhibit  everywhere  the  same  interference-color.  There- 
fore the  appearance  of  the  same  color  shade  throughout  the 
whole  extent  of  the  plate  is  a  very  delicate  means  of  checking 
the  uniformity  of  its  thickness. 

But  different  polarization-colors  at  different  points  are  ex- 
hibited also  by  a  plate  whose  thickness  is  everywhere  the  same, 
if  it  is  cut  not  from  a  homogeneous  crystal  but,  for  example, 
from  an  aggregate  of  crystals  of  the  same  kind  intergrown  with 
one  another  in  different  orientations;  because,  as  will  be  shown 
in  the  following,  in  all  doubly  refracting  crystals  the  strength 
of  the  double  refraction  varies  with  the  direction.  Now  many 
so-called  "  compact  "  minerals  are  crystalline  aggregates,  i.e. 
consist  of  very  small  crystalline  particles  oriented  irregularly 
with  respect  to  one  another;  and  hence,  if  these  particles  are 
doubly  refractive,  a  very  thin  plate  prepared  from  the  aggre- 
gate, viewed  between  two  crossed  Nicol  prisms,  exhibits  aggre- 
gate polarization:  the  single  particles  differ  from  one  another  in 
their  coloring  and  in  their  position  of  extinction  on  rotation  of 
the  plate.  Thereby  the  texture  of  such  an  aggregate  is  brought 
into  bold  relief,  while  without  the  use  of  polarized  light  the  ag- 


72  DOUBLE    REFRACTION   OF   LIGHT 

gregate  may  possibly  appear  quite  uniform.  We  have  to  do  here 
with  a  body  that  is  not  homogeneous-,  and  this  is  sometimes  the 
case  also  with  freely  developed  crystals.  By  disturbances  in 
the  growth  of  one  of  these  it  may  happen  that  single  parts  of  it 
have  been  deflected,  as  it  were,  from  their  normal  orientation. 
Since  the  vibration  directions  of  light  in  doubly  refracting 
crystalline  bodies  stand  in  a  definite  relation  to  the  arrange- 
ment of  the  smallest  particles  of  the  same,  the  deflection  of  any 
part  of  such  a  body  must  be  accompanied  by  a  variation  of 
these  vibration  directions,  and  between  crossed  nicols  the  parts 
in  question  do  not  then  extinguish  in  the  same  positions  as  does 
the  rest  of  the  crystal.  Again,  if  during  the  growth  of  a  crystal 
a  gradual  change  in  the  chemical  composition  has  taken  place, 
so  that  in  their  nature  the  outer  zones  of  the  crystal  differ 
the  most  from  the  inner,  and  if  to  this  difference  there  corre- 
sponds at  the  same  time  a  difference  in  the  orientation  of  the 
vibration  directions,  then  the  crystal  does  not  appear  dark  in 
its  entire  extent  at  once,  but  on  rotation  the  extinction  travels 
over  it. 

In  such  cases  it  is  a  matter  of  aggregates  of  crystals  not 
parallel,  or  else  of  conglomerates  of  crystals  of  different  kinds. 
It  not  infrequently  happens  in  the  solidification  of  molten  metals, 
in  the  slow  congelation  both  of  artificial  vitreous  fluxes  and 
of  natural  eruptive  rocks,  and  finally  sometimes  also  during 
the  crystallization  from  solutions,  that  crystal  aggregations  of 
spherical  shape  are  formed  which  have,  within,  a  radiating- 
fibrous  structure  such  that  the  single  fibers  meeting  at  the 
center  of  the  sphere  each  consist  of  one  crystal.  Now  if  from 
such  a  mass,  a  so-called  spherulite,  one  cuts  a  very  thin  central 
plate,  and  if  the  vibration  directions  of  the  radial  fibers  com- 
posing the  plate  are  parallel  and  perpendicular  respectively  to 
the  long  direction  of  these  fibers,  then  in  parallel  light  between 
crossed  nicols  such  a  plate  exhibits  a  dark  cross,  whose  arms 
are  parallel  to  the  principal  sections  of  the  nicols.  The  expla- 
nation of  this  phenomenon  is  very  simple:  The  fibers  whose 


POLARIZATION-COLORS  73 

long  direction  passes  parallel  to  the  polarization  plane  of  either 
nicol  must,  since  their  vibration  directions  are  parallel  and  perpen- 
dicular to  the  long  direction,  obviously  appear  dark;  and  the 
immediately  adjacent  fibers  nearly  dark,  since  their  vibration 
directions  diverge  but  little  from  those  of  the  nicols;  yet,  the 
greater  the  angle  that  a  fiber  makes  with  these  latter  directions, 
the  more  is  it  lighted  up  by  the  polarized  light,  and  this  is  the 
case  in  the  highest  degree  with  the  fibers  standing  at  45°  to  the 
nicols.  Consequently,  from  the  center  of  the  radiating-fibrous 
mass  there  must  pass  out  four  black  arms,  becoming  broader 
outward;  and  these  arms  divide  the  surface  into  four  quadrants, 
whose  brightness  is  maximum  on  the  intermediate  radius, 
diminishing  toward  the  dark  arms.  Since  spherulites  are  often 
so  small  that  even  with  microscopical  observation  of  the  sections 
their  fibrous  structure  can  not  be  perceived,  the  described 
phenomenon,  which  with  such  a  body  always  appears  when 
the  nicols  are  crossed,  may  serve  to  distinguish  the  radiating- 
fibrous  character  of  a  spherulite.  This  interference  phenom- 
enon may  be  very  easily  imitated  if  a  crystal  extinguishing 
parallel  to  its  long  direction  is  cemented  over  a  radial  opening 
cut  in  an  opaque  disk :  when  the  disk  is  rapidly  rotated  between 
two  crossed  nicols  the  crystal  takes  up  in  quick  succession  the 
different  positions  of  the  single  fibers  of  a  radiating-fibrous 
plate. 

In  explaining  the  above-described  phenomenon  it  has  been 
supposed  that  we  have  to  do  with  a  plate  passing  through  the 
center  of  the  spherulite,  and  so  thin  that  it  consists  only  of  a 
layer  of  radial  fibers  lying  parallel  to  its  plane.  If  the  plate  is 
thicker,  and  if  therefore  only  one  or  neither  of  its  two  faces 
passes  through  the  center  of  the  spherulite,  then  at  its  center 
the  preparation  consists  of  fibers  standing  perpendicular,  while 
around  this  central  axis  it  consists  of  inclined  fibers,  whose 
inclination  increases  with  their  distance  from  the  center.  In 
these  inclined  fibers  therefore  the  rays  experience  a  different 
double  refraction,  and  consequently  from  the  center  of  the 


74  DOUBLE    REFRACTION    OF   LIGHT 

plate  out  to  the  edge  there  appears  a  succession  of  different  in- 
terference-colors. When  the  vibration  directions  of  the  fibers 
composing  the  spherulite  are  oblique  to  their  long  direction, 
the  arising  cross,  also,  lies  oblique  to  the  vibration  directions 
of  the  nicols. 

As  Biitschli  has  recently  shown,  a  similar  interference 
phenomenon  (a  cross  parallel  to  the  vibration  directions  of 
the  nicols)  arises  in  amorphous  bodies, — e.g.  in  flakes  of 
resin — by  reason  of  concentric,  slanting  cracks,  and  in  con- 
sequence of  the  polarization  effected  at  these  cracks  by  total 
reflection.  (Cf.  p.  49.) 

POLARIZATION  APPARATUS 

For  observing  the  phenomena  brought  about  by  the  double 
refraction  of  light,  so  far  as  they  have  yet  been  considered,  ser- 
vice is  made  of  the  polarization  apparatus  shown  in  vertical 
section  in  Fig.  34.  This  consists  of  a  mirror,  m,  so  inclined  that 
by  it  the  light  coming  from  a  bright  part  of  the  sky  is  reflected 
vertically  upward;  and  of  a  polarizing  nicol,  p,  which  with  ad- 
vantage is  put  in  a  tube,  between  two  glass  lenses,  /  and  /,  in 
such  a  way  that  the  common  focus  of  these  lenses  falls  in  the 
center  of  the  nicol.  Thus  the  rays  of  light,  falling  parallel  upon 
the  first  lens,  must  all  pass  through  the  nicol  drawn  together 
into  the  form  of  a  double  cone,  and  hence  emerge,  again  par- 
allel, from  the  second  lens.  The  instrument  consists  further  of 
the  stage,  5,  with  a  horizontal,  rotatable  glass  plate  on  which 
is  laid  the  crystal  to  be  investigated,  k\  and  finally  of  the  ana- 
lyzing nicol,  a,  which  brings  the  doubly  refracted  vibrations  back 
to  one  plane  of  polarization.  Both  nicols,  as  well  as  the  stage, 
are  rotatable  about  the  vertical  axis  of  the  apparatus.  The 
polarizer  may  be  replaced  by  substituting  for  the  reflecting  mir- 
ror a  bundle  of  thin  glass  plates,  which  according  to  page  49 
gives  to  the  reflected  light  a  certain  degree  of  polarization,  al- 
though not  a  quite  complete  one.  But  the  use  of  such  a  polar- 
izer is  frequent,  because  it  is  far  cheaper  to  manufacture  than 


POLARIZATION   APPARATUS 


75 


I  l\al 


is  a  Nicol  prism  of  so  large  a  size  as  is  desirable  for  the 
illuminating  power  of  the  instrument.  Finally,  the  simplest  — 
and,  to  be  sure,  also  the  least  perfect 
—  form  of  polarization  apparatus  is 
the  so-called  tourmaline  tongs,  con- 
sisting of  two  tourmaline  plates 
(cf.  p.  54)  so  mounted  as  to  be 
rotatable  with  reference  to  each 
other;  the  crystal  to  be  investigated 
is  simply  wedged  in  between  the  two 
plates  and  observed  by  directing  the 
little  instrument  toward  the  bright 
sky. 

The  polarization  apparatus  pic- 
tured in  Fig.  34  is  called  also  an 
orthoscope,  because,  with  it,  only 
those  light  rays  that  pass  through 
the  crystal  plate  at  right  angles  are 
investigated  for  the  alteration  they 
have  suffered  in  the  crystal.  When 
the  crystal  is  not  visible  to  the  naked  i 
eye,  i.e.  if  to  observe  it  a  microscope 
is  requisite,  the  microscope  is  converted  into  an  orthoscope  by 
inserting  a  Nicol  prism  below  the  stage  — which  for  such  in- 
vestigations must  of  course  be  rotatable  —  and  mounting  a 
second  nicol  on  the  eyepiece  or  putting  it  between  the  eyepiece 
and  the  objective,  within  the  microscope  itself.  [The  instru- 
ment is  then  known  as  a  polarization-microscope.]  Now  if  we 
look  into  such  an  instrument  when  the  upper  nicol  is  crossed 
as  regards  the  lower,  the  field  of  view  appears  dark,  and  so  do 
all  singly  refracting  bodies  that  may  be  in  it.  But  if  we  place 
on  the  stage  a  transparent  preparation  containing  doubly  re- 
fractive crystals,  e.g.  a  so-called  "  thin  section  "  of  a  rock  (i.e. 
a  plate  ground  extremely  thin),  then,  when  the  stage  is  rotated 
in  its  own  plane  these  crystals  must  pass  through  the  change 


Fig.  34- 


j6  DOUBLE   REFRACTION   OF  LIGHT 

from  dark  to  light  or  colored  elucidated  in  the  foregoing  sec- 
tions. If  the  mounting  of  the  rotatable  nicols  is  provided  with 
marks  by  whose  aid  the  nicols  may  at  any  time  be  given  a 
definite  orientation  in  which  they  are  at  the  same  time  crossed, 
and  if  a  pair  of  cross-hairs  is  brought  into  the  focal  plane  of  the 
microscope  in  such  a  way  that  one  of  the  two  hairs  then  appear- 
ing in  the  field  has  exactly  the  vibration  direction  of  the  light 
emerging  from  the  lower  nicol  and  the  other  hair  the  direction 
of  the  vibrations  transmitted  by  the  analyzer,  then  does  the  in- 
strument permit  one  not  only  to  determine  the  existence  and 
the  strength  of  the  double  refraction  of  a  crystal,  by  means 
of  the  color  appearing  on  rotation,  according  to  page  63  et  seq.y 
but  also  to  ascertain  the  vibration  directions  of  the  two  rays 
arising  in  the  crystal;  for,  according  to  page  61,  these  directions 
are  given  by  the  positions  of  darkness  of  the  crystal,  provided 
the  vibration  directions  of  the  nicols  are  known. 

For  determining  the  vibration  directions  of  a  crystal  the  edge 
of  the  rotatable  stage  is  provided  with  a  graduated  circle,  so 
that  its  every  position  can  be  read  off  at  a  fixed  mark.  After 
first  bringing  the  crystal  to  be  investigated  into  the  middle  of 
the  field,  one  rotates  the  stage  until  a  straight  boundary  of  the 
crystal  touches  or  is  parallel  to  one  of  the  cross-hairs  throughout 
its  length,  and  reads  this  position  of  the  stage  on  the  graduated 
circle.  In  general  neither  vibration  direction  will  be  parallel  to 
this  boundary  of  the  crystal  plate,  and  the  plate  will  therefore 
not  appear  dark;  but  if  one  rotates  the  stage,  and  with  it  the 
crystal,  from  the  position  first  read  to  that  where  extinction  of 
the  light  occurs  (in  which  position  the  vibration  directions  of  the 
crystal  coincide  with  those  of  the  nicols),  then  the  rotation,  read 
off  on  the  graduated  circle,  gives  the  extinction  angle,  — i.e.  the 
angle  formed  by  one  of  these  vibration  directions  with  the  pre- 
viously adjusted  crystal  boundary.* 

Both  with  the  polariscope  (cf.  p.  59)  described  on  page  74 

*  For  particulars  as  to  the  arrangement  and  use  of  polarization-microscopes 
cf.  footnote  f,  P-  3°- 


POLARIZATION   APPARATUS 


77 


and  with    the  microscope   arranged    for  polarization,   we  can, 
in  general,  investigate  only  those  optical  phenomena  that  the 
crystal  exhibits  in  one  direction.     Therefore  investigations  of 
this  kind  are  termed  investigations  in 
parallel  polarized  light. 

But  very  frequently  one  aims  to  ob- 
serve at  once  the  alterations  which 
the  polarized  light  suffers  in  passing 
through  a  crystal  along  directions  as 
different  as  possible.  This  purpose  is 
served  by  the  polariscope  for  obser- 
vations in  convergent  light,  called 
also  a  conoscope.  This  instrument, 
represented  in  Fig.  35  in  vertical  sec- 
tion through  the  axis,  consists  of  a 
mirror,  m,  and  of  the  polarizer,  p, 
which  is  enclosed  between  two  lenses, 
/  and  /,  exactly  as  with  the  orthoscope. 
Behind  the  upper  lens  /  is  a  screen 
(diaphragm)  of  blackened  metal  hav- 
ing in  it  a  circular  opening  of  the 
diameter  de.  This  opening  is  illumi- 
nated from  the  bright  sky  through  the 
medium  of  the  mirror  m,  the  nicol 
p,  and  the  lenses  /;  every  point  of  it  by 
a  cone  of  rays  whose  fulcrum  is  that 
point  itself  and  whose  base  is  the  lower 
lens  /.  In  the  figure  these  ray  cones  are  / 
given  for  the  two  points  d  and  e,  at 
the  edge  of  the  diaphragm,  and  for 
its  center,  c;  but  below  the  upper  lens  /  the  cones  are  continued 
only  in  dots,  because,  on  account  of  the  refraction  in  the  lenses, 
the  true  path  of  the  rays  is  other  than  that  indicated:  their  path 
is,  indeed,  such  that  all  rays  falling  parallel  on  the  lower  lens 
are  again  parallel  after  their  exit  from  the  upper,  but  naturally 


Fig-  35- 


DOUBLE    REFRACTION    OF    LIGHT 


then  become  components  not  of  the  same  cone  but  of  different 
ones  of  the  cones  of  light  falling  respectively  on  c,  d,  e,  etc. 
Hence,  it  is  the  bright  opening  de,  illuminated  from  below  by 

plane-polarized  light,  that  we  look 
at  through  this  instrument.  We 
may  therefore  consider  every  point 
of  this  opening  —  e.g.  the  point  d  — 
as  a  point  from  which  diverging 
light  rays  proceed;  these  rays  do 
not,  however,  proceed  in  all  direc- 
tions, as  from  a  self-luminous  point, 
but  only  in  suchdirectionsas  lie  within 
the  conical  angle  of  the  cone  whose 
fulcrum  is  d  and  whose  base  is  the 
lower  lens  /.  The  rays  proceeding 
from  d  are  rectilinear  prolongations 
of  those  of  the  cone  in  question. 
If  we  follow  their  path  upward,  we 
see  these  diverging  rays  impinge  on 
a  very  convex  lens,  n\  this  stands, 
from  the  diaphragm  de,  at  exactly  the 
distance  of  its  focal  length,  so  that  in 
consequence  all  the  rays  diverging 
from  a  point  of  the  focal  plane  de  are 
transformed  by  n  into  parallel  rays. 
Above  n  there  is  a  lens  0,  of  the 
same  size  and  convexity,  mounted  at 
the  lower  end  of  a  vertically  movable 
tube,  which  contains  also  the  eye- 
piece, <?;  this  tube  is  depressed  until 


Fig.  35- 


the  focus  of  o  and  that  of  n,  which  has  the  same  focal  length  as  0, 
coincide  at  /.  The  ray  cone  proceeding  from  d  is  refracted  at 
the  left  side  of  n  and  transformed  thereby  into  a  ray  cylinder;  this 
cylinder  passes  through  the  right  side  of  o  and,  since  the  rays  are 
parallel,  is  again  concentrated  in  the  focal  plane  of  o  at  the  point 


POLARIZATION    APPARATUS  79 

d,  corresponding  to  d.  In  the  figure  the  same  construction  is  car- 
ried out  for  the  rays  that  proceed  from  the  center  c  of  the  illu- 
minated opening  de,  which  must  converge  at  ?-;  also  for  those 
that,  coming  from  e,  converge  at  e.  Since  the  same  applies  to  all 
points  of  the  opening  de,  an  image  of  this  opening  must  arise  in 
the  plane  de.  This  image,  then,  we  observe  with  a  magnifying 
glass  of  very  low  power;  namely,  with  the  eyepiece  q,  through 
which  we  see  a  so-called  virtual  image  approximately  in  the  plane 
£V.  The  rays  that  seem  to  come  from  this  image  pass,  before  they 
arrive  in  our  eye,  through  the  analyzing  nicol,  a.  Hence,  if  upon 
the  stage,  s,  we  lay  a  plane-parallel  plate  of  a  crystallized  medium 
(represented  in  the  figure  by  the  dotted  outline)  in  such  a  way 
that  /  falls  within  the  plate,  the  plate  is  traversed  by  ray  sys- 
tems of  very  diverse  direction;  all  rays  of  the  same  direction  — 
which  in  a  homogeneous  crystal  must  of  course  experience  the 
same  alterations  — converge  at  a  single  point  of  the  image  £V,  all 
those  of  other  directions  at  other  points.  Thus,  in  the  image  d'e' 
we  are  able  to  view  at  a  glance  all  the  interference  phenomena 
suffered  within  the  crystal  to  be  investigated  by  rays  of  very 
many  diverse  directions,  —  namely,  of  all  the  directions  that  lie 
within  the  cone  projected  from  /  upon  the  circumference  of  the 
lens  n,  —  and  we  then  see  in  the  field  of  view  a  so-called  inter- 
ference-figure *  of  the  crystal,  different  parts  of  which  figure, 
corresponding  to  different  directions  of  the  rays  within  the 
crystal,  exhibit  different  color  and  different  light  intensity.  The 
shorter  the  focal  distance  of  n  and  o,  the  greater  is  the  conical 
angle  of  the  cone  projected  on  them  from  the  focus,  and  the 
larger  is  the  field  of  the  instrument.  Since  it  is  often  necessary 
to  concentrate  within  the  field  rays  that  have  passed  through 
the  crystal  in  very  divergent  directions,  Norrenberg  replaced 
each  of  the  two  lenses  n  and  o  with  a  system  of  several  plano- 

*  As  follows  from  Fig.  35,  the  rays  inclined  to  the  right  converge  on  the  right 
side  of  the  field;  so  the  interference-figure  exhibits  the  phenomena  on  the  same 
side  to  which  the  rays  are  inclined  in  the  crystal,  while  the  image  of  the  crystal 
itself,  which  one  sees  on  removing  the  tube,  is  reversed. 


8O  DOUBLE   REFRACTION   OF  LIGHT 

convex  lenses  almost  touching  one  another,  — these  lenses  hav- 
ing, together,  the  effect  of  a  lens  of  very  short  focus.  Constructed 
in  this  form,  which  is  now  in  common  use,  the  apparatus  is  there- 
fore often  designated  as  "  Norrenberg's  polariscope  ". 

The  apparatus  just  described  serves  for  investigating  fairly 
large  crystal  plates  in  convergent  light,  exactly  as  the  one  pic- 
tured in  Fig.  34  serves  for  investigating  them  in  parallel  light. 
But  just  as  a  microscope  with  rotatable  stage  can,  through  the 
addition  of  two  Nicol  prisms,  be  made  available  for  the  ortho- 
scopic-optical  investigation  of  small  crystals,  so  also  may  it  be 
transformed  into  a  conoscope,  with  which  even  a  crystal  of 
microscopic  smallness  can  be  investigated  in  convergent  polar- 
ized light.  To  this  end  it  is  necessary,  besides  adding  the 
two  nicols,  only  to  insert  a  condensing  lens  into  the  opening  in 
the  stage,  directly  under  the  crystal,  — which  lens  projects 
strongly  convergent  light  rays  through  the  crystal  lying  at  the 
center  of  the  field, —  and  to  adjust  the  optical  parts  of  the 
microscope  proper  no  longer  on  the  crystal,  but  on  the  inter- 
ference-figure arising  in  another  plane.  The  latter  purpose  is 
accomplished  either  by  removing  the  eyepiece,  after  which  one 
sees  the  very  small  interference-figure  at  a  definite  place  in  the 
in  tube;  or  by  inserting  an  auxiliary  eyepiece,  which  corrects 
the  adjustment  and  at  the  same  time  magnifies  the  interference- 
figure,  so  that  in  the  field  one  then  sees  not  the  crystal,  but  the 
interference-figure  produced  by  it.  *  f 

*  For  the  more  detailed  description  of  conoscopes  and  the  methods  of  using 
them  cf.  footnote  f,  P-  30- 

f  [As  regards  application  of  the  polarization-microscope  to  petrographical 
investigations  see  especially  the  following:  Joseph  P.  Iddings:  "Rock  Minerals", 
New  York,  1904;  Rosenbusch-Iddings:  "Microscopic  Physiography  of  the 
Rock-making  Minerals",  New  York,  1888;  N.  H.  and  Alexander  Winchell: 
"Elements  of  Optical  Mineralogy",  New  York,  1909;  L.  M.  I.  Luquer: 
New  York,  1908.] 


DOUBLE    REFRACTION    OF   LIGHT    IN   CALCITE 


8l 


OPTICALLY  UNIAXIAL  CRYSTALS 
DOUBLE    REFRACTION    OF    LIGHT    IN    CALCITE 

The  relations  in  which,  in  a  doubly  refracting  crystal,  the 
vibration  directions  and  transmission  velocities  of  the  two  rays 
arising  by  the  double  refraction  stand  to  the  form  of  the  crystal, 
were  perceived  first  in  calcite  (crystallized  calcium  carbonate), 
and  this  mineral  thereby  became  the  point  of  departure  in  the 
investigation  of  the  laws  of  double  refraction.  Calcite  cleaves 
very  perfectly  along  three  directions  forming  with  one  another 


d 

Fig-  36-  Fig.  37. 

angles  of  74°  56',  and  since  it  occurs  in  nature  (especially  in 
Iceland)  in  very  large  crystals  of  watery  clearness,  large,  trans- 
parent cleavage  blocks  can  be  easily  produced  which  have  the 
form  of  a  so-called  rhombohedron.  (See  Fig.  36.)  These  blocks 
are  highly  suitable  for  the  study  of  double-refraction  phenomena, 
especially  because  calcite  has  the  property  of  deviating  the  two 
rays  transmitted  in  it  an  amount  whose  difference  for  the  two 
rays  is  very  considerable.  Let  Fig.  37  represent  a  so-called 
principal  section  of  the  rhombohedron;  i.e.  a  section  through  the 
four  solid  angles,  a,  b,  c,  d  (Fig.  36),  so  that  ab  and  cd  are  the 
short  diagonals  of  the  rhombohedron  faces.  The  plane  of  such 
a  principal  section  divides  the  rhombohedron  symmetrically,  i.e. 
into  two  equal  and  opposite  halves;  it  is  therefore  said  to  be  a 
plane  of  symmetry  of  the  rhombohedron.  Now  if  upon  one  of 
the  rhombohedron  faces,  as  ab  in  Fig.  37,  we  cause  to  fall  per- 
pendicularly a  bundle  of  rays  of  ordinary  light  that  are  parallel  to 


82  OPTICALLY   UNI  AXIAL   CRYSTALS 

the  line  mn  lying  in  the  principal  section,  for  example,  if  we 
look  through  the  two  parallel  rhombohedron  faces  toward  a 
small,  round,  illuminated  opening  in  a  dark  screen,  we  see  two 
equally  bright  images  of  that  opening.  Whence  it  follows  that 
a  light  ray  entering  the  calcite  along  the  line  mn  is  split  up  into 
two  rays,  of  which  one,  no,  like  a  ray  of  ordinary  light  with  per- 
pendicular incidence,  passes  through  the  calcite  unrefracted, 
while  the  second,  deflected  in  the  plane  of  the  principal  section, 
moves  in  the  crystal  from  n  to  p  and  there  suffers  the  opposite 
refraction  to  that  at  n,  emerging  therefore  along  the  direc- 
tion pe.  Since,  of  these  two  rays,  only  the  first,  no,  conforms 
to  the  law  of  refraction  for  ordinary  light,  it  is  called  the  ordi- 
nary ray  (0),  the  other  the  extraordinary  (e). 

If  the  calcite  is  made  rotatable,  in  a  mounting,  about  an 
axis  parallel  to  the  direction  of  the  incident  light,  then,  of  the 
two  images  now  visible  of  the  bright  opening  in  the  dark  screen 
set  in  front  of  the  rhombohedron,  it  can  easily  be  determined 
which  is  the  ordinary  and  which  the  extraordinary;  for  on 
rotation  of  the  rhombohedron  the  ordinary  image,  not  being 
deflected,  remains  stationary,  while  the  extraordinary,  deflected 
in  the  plane  of  the  principal  section,  moves  around  the  ordi- 
nary image  in  a  circle.  If  into  absolutely  parallel  position  be- 
fore this  calcite  we  now  bring  a  second,  of  equal  size  and 
mounted  in  just  the  same  way,  then  on  looking  through  both 
calcites  toward  the  bright  opening  we  see,  first,  that  the  ordinary 
image  o  emerging  from  the  first  calcite  passes  through  calcite  // 
wholly  as  an  ordinary  image,  without  being  again  doubly  re- 
fracted; and  second,  that  just  as  little  doubly  refracted  is  the 
extraordinary  image  e,  which,  unlike  o,  passes  through  calcite 
//  as  an  extraordinary  image,  since  it  suffers  a  further  deflection 
of  the  same  amount  as  in  /:  we  see,  as  before,  two  equally  bright 
images,  one  above  the  other,  but  at  double  the  distance.  If 
we  now  rotate  the  foremost  calcite  to  the  right  or  left  in  its 
mounting,  there  appear  four  images  of  the  opening,  but  of 
different  brightness;  and  from  the  deflections  it  is  easily  under- 


DOUBLE   REFRACTION   OF   LIGHT   IN   CALCITE  83 

stood  that  each  of  the  two  previous  images  o  and  e  has  been 
resolved  into  an  ordinary  and  an  extraordinary.  Figure  38  shows 
the  phenomenon  observed 
when  the  principal  section 
of  calcite  //,  that  next  to  the 
eye,  has  been  rotated  a  small 
angle  to  the  right.  While 
with  this  calcite  in  parallel 
position  to  the  first,  /,  there 
were  visible  only  an  ordinary 
image  o  and  an  equally 
bright  extraordinary  image 
in  the  position  e2,  the  image 
e1  (by  el  is  indicated  the 

position  of  the  extraordinary  image  when  only  the   first  calcite 

stands  before  the  bright 
opening)  now  suffers  a  de- 
flection in  the  principal  sec- 
tion of  the  second  calcite, 
appearing  therefore  at  se,  but 
with  somewhat  less  bright- 
ness; and  at  ev  i.e.  without 
deflection,  a  second  faint 
image  appears:  on  the  other 
hand,  o  too  has  diminished  in 
brightness,  and  there  now  ap- 
pears, deflected  in  the  prin- 
cipal section  of  the  second  calcite, —  with  extraordinary  refrac- 
tion, therefore  — an  image  sw,  of  the  same  slight  brightness  as 
the  now  ordinary  er  The  farther  we  rotate,  the  more  does  the 
brightness  diminish  of  the  ordinary  image  o,  formed  from  the 
first  ordinary,  and  of  the  extraordinary  image  se,  arising  from 
the  extraordinary,  while  the  brightness  of  the  other  two  images 
continuously  increases;  with  45°  rotation  all  four  images  are 
equally  bright  and  are  in  the  position  represented  in  Fig.  39. 


84 


OPTICALLY   UNIAXIAL   CRYSTALS 


Fig.  40. 


When  the  principal  sections  of  the  two  calcites  stand  mutually 
perpendicular  the  ordinary  image  of  o  and  the  extraordinary 

of  e  entirely  vanish  :  there  ap- 
pear only  two  images,  £0  and 
ev  —  the  deflected,  hence  ex- 
traordinary, of  o,  and  the 
not  further  refracted,  hence 
ordinary,  of  e.  (See  Fig.  40.) 
After  a  rotation  of  180°  there 
appears  only  a  single  image, 
since  in  the  second  calcite 
e  is  deflected  the  same 
amount  as  in  the  first  but 
in  the  opposite  direction, 
consequently  being  after  its 
exit  identical  with  the  image 
o.  To  sum  up:  During  one 

whole  rotation  of  one  of  the  two  calcites  through  360°  there  are, 
as  observation  shows,  only  four  positions  (viz.  when  the  prin- 
cipal sections  of  the  two  calcites  intersect  at  45°),  when  the 
rays  o  and  e  emerging  from  the  first  rhombohedron  behave  like 
rays  of  ordinary  light,  i.e.  each  become  resolved  into  two  rays 
of  equal  brightness:  in  all  other  positions  there  arise  from  each 
of  them  two  rays  of  unequal  intensity;  except,  finally,  with 
the  principal  sections  mutually  parallel  or  perpendicular,  when 
the  two  rays  are  not  doubly  retracted  at  all.*  f 

From  this  behavior,  therefore,  it  is  seen  that  each  of  the  two 
rays  (really  bundles  of  rays)  emerging  from  the  first  calcite 
exhibits  a  complete  symmetrical  one-sidedness  with  reference  to 

*  These  phenomena  are  suitable  for  distinguishing  plane-polarized  light  from 
ordinary  light,  somethmg  of  which  the  eye  is  incapable. 

f  [For  an  interesting  discussion  of  the  appearance  of  a  bright  spot  viewed  through 
calcite  (also  through  other  uniaxial,  as  well  as  through  singly  refracting  and  biaxial 
crystals),  showing  how  these  phenomena  are  modified  (the  images  distorted,  etc.), 
see  a  posthumous  article  by  the  late  H.  C.  Sorby:  "The  Optical  Properties  of 
Crystals",  Min.  Mag.  1909,  16,  189-215.] 


DOUBLE   REFRACTION   OF   LIGHT  IN   CALCITE  85 

one  of  the  diagonals  of  the  rhombus  that  represents  the  surfaces 
of  entrance  and  emergence  of  the  light, — to  that  diagonal, 
namely,  that  corresponds  to  the  principal  section.  If,  there- 
fore, we  would  explain  these  phenomena  by  the  arising  in  the 
crystal  of  two  rays  polarized  perpendicularly  to  each  other,  we 
can  do  so  in  no  other  way  than  by  assuming  that  the  vibration 
direction  of  one  of  these  rays  coincides  with  the  diagonal  men- 
tioned, and  that  the  vibration  direction  of  the  other  ray  coin- 
cides with  the  second  diagonal  (the  longer  one  of  the  rhombus), 
perpendicular  to  the  diagonal  first  mentioned. 

Assumed:  that  the  ordinary  ray  o  emerging  from  the  calcite 
consists  only  of  rectilinear  vibrations  parallel  to  the  longer  diago- 
nal of  the  rhombohedron  face,  the  extraordinary  e  only  of  such 
as  are  parallel  to  the  shorter  diagonal.  If  I  and  //  in  Fig.  41 


Fig.  41. 

might  represent,  side  by  side,  the  two  calcite  rhombohedrons 
through  which  the  light  is  to  pass,  — which  therefore  are  to  be 
imagined  as  lying  one  before  the  other, —  then  would  oo  and  o'o' 
be  the  vibration  directions  in  them  of  the  ordinary  ray  whose 
transmission  direction  stands  perpendicular  to  the  plane  of  the 
figure,  ee  and  e'e'  those  of  the  extraordinary  ray  of  that  direc- 
tion. The  ordinary  ray  emerging  from  /  with  the  vibration 
direction  oo  arrives  in  //,  when  /  and  //  are  parallel,  as  Fig.  41 
represents,  with  the  vibration  direction  o'o',  and  the  motion  is 
here  resolved  into  one  along  o'o'  and  one  along  e'e';  the  portion 
falling  to  the  latter  direction  is  zero,  however,  while  the  compo- 
nent parallel  to  o'o'  corresponds  to  the  total  intensity  of  the 
ordinary  ray  entering  //;  so  in  the  second  calcite  this  ray  is 


86 


OPTICALLY   UNIAXIAL   CRYSTALS 


not  doubly  refracted,  but  passes  through  wholly  as  an  ordinary 
ray.  If  we  now  imagine  //  as  rotated  a  small  angle  (Fig. 
42),  the  ordinary  ray  from  7  enters  77  with  the  vibration 
direction  wco,  being  here  resolved  into  two  rays  which  vibrate 
parallel  to  o'o'  and  e'e* ',  and  whose  amplitudes  are  proportional 
to  the  lengths  CO  and  CE.  In  other  words,  the  ray  does  not 


e'  e 


Fig.  42. 


pass  through  with  its  full  intensity  as  an  ordinary  ray:  the  re- 
mainder is  transmitted  as  an  extraordinary  ray,  — deflected, 
therefore,  in  the  principal  section  of  //,  as  shown  in  Fig.  38. 
So  in  this  case  the  ordinary  ray  emerging  from  the  first  calcite  is 
in  the  second  doubly  refracted,  yielding  however  two  rays  of  differ- 
ent intensity.  With  a  further  rotation  of  calcite  II  (Fig.  43), 


«<-4— -*—  -4-; 


Fig.  43- 

namely,  when  it  forms  45°  with  7,  the  ordinary  ray  is  by  77 
resolved  into  two  components  of  equal  magnitude,  thus  yielding 
two  rays,  an  ordinary  and  an  extraordinary  of  equal  brightness; 
accordingly  it  passes  through  the  second  calcite  only  to  the  half 
part  as  an  ordinary  ray.  This  is  exactly  the  phenomenon 
shown  in  Fig.  39.  Now  the  farther  one  rotates  77,  the  smaller 
is  the  portion  of  the  horizontal  ether  vibrations  emerging  from  7 
that  passes  through  II  as  an  ordinary  ray;  and  when  finally 


DOUBLE   REFRACTION   OF   LIGHT   IN   CALCITE 


the  calcites  have  the  position  represented  in  Fig.  44,  in  which 
position  /  and  //  are  crossed  at  right  angles,  the  ordinary  ray 
coming  out  of  /  enters  //  with  the  vibration  direction  e'e',  being 
resolved  therefore  into  two  mutually  perpendicular  components 
of  which  the  one,  parallel  to  0V ',  is  zero,  while  the  other,  parallel 
to  e'e',  is  equal  to  the  whole  amplitude  of  the  entering  light.  So 
with  this  position  of  the  calcites  the  ray  o  can  nowise  pass  through 
the  second  calcite  as  an  ordinary  ray:  it  must  go  through  wholly 
as  an  extraordinary.  (Cf.  Fig.  40.)  In  part  again  as  an  ordi- 
nary ray  does  the  ray  o  pass  through  II  on  further  rotation  of 
the  latter,  wholly  as  such  when  the  total  rotation  amounts  to 
1 80°; — and  it  is  evident  that  entirely  the  same  would  occur 

T  ii 


Fig.  44. 


o' 


if  the  rotation,  starting  out  from  the  position  in  Fig.  41, 
should  take  place  to  the  left,  instead  of  to  the  right.  The  fun- 
damental assumption  as  to  the  nature  of  the  light  that  has 
passed  through  the  first  calcite  accordingly  teaches,  just  as  do 
the  observed  and  previously  described  phenomena,  that  the 
principal  section  of  the  calcite  rhombohedron  is  a  distinctive  char- 
acter of  the  ordinary  ray  emerging  from  the  first  rhombohedron, 
inasmuch  as  this  ray  passes  through  //  wholly  as  an  ordinary  ray 
when  the  principal  section  of  //  is  parallel  to  that  of  /,  the  less 
as  an  ordinary  ray  the  greater  the  angle  included  between  the  two 
principal  sections,  and  finally  not  at  all  as  such  when  the  prin- 
cipal sections  stand  mutually  perpendicular.  So  this  plane  which 
is  thus  characteristic  of  the  ray  o  is  the  plane  that  stands  per- 
pendicular to  the  vibration  direction  assumed  for  this  ray;  in 
other  words,  it  is  the  plane  designated  on  page  15  as  the  plane  oj 


88 


OPTICALLY   UNIAXIAL   CRYSTALS 


polarization  of  a  plane-polarized  light  ray.  Therefore  we  say 
the  ordinary  ray  is  polarized  in  the  principal  section  of  the  crystal. 
Let  us  now,  under  the  same  assumptions,  consider  the  second, 
the  extraordinary,  light  ray  emerging  from  calcite  /.  Figure  41 
or  Fig.  45  shows  that  this  ray,  entering  //  with  the  vibration 


Fig.  45- 

direction  e'e',  is  resolved  into  two  components  parallel  to  o'o' 
and  e'e'  respectively,  of  which  components  the  former  is  zero; 
but  this  component  has  the  vibration  direction  of  the  ordinary 
ray;  and  accordingly,  when  the  calcites  are  parallel,  the  ray  e 
from  /  can  nowise  pass  through  //  as  an  ordinary  ray,  as  is 


Fig.  46. 

taught  in  fact  by  observation.  (Cf.  p.  82.)  If  we  rotate  // 
somewhat  (see  Fig.  46),  then  e  enters  with  the  vibration  direc- 
tion 6£,  and  is  resolved  into  the  two  components  CO'  and  CE'. 
Thus  the  extraordinary  ray  from  the  first  calcite  now  passes  in 
part  as  an  ordinary  through  the  second ;  and  moreover  in  greater 
part,  the  farther  we  rotate.  When  finally,  as  in  Fig.  47,  the 
plane  perpendicular  to  the  principal  section  of  the  second  cal- 
cite —  which  plane  is  determined  by  o'o'  and  the  transmission 
line  of  the  ray  —  stands  parallel  to  the  principal  section  of  the 
first  calcite,  the  extraordinary  ray  emerging  from  this  latter 


DOUBLE   REFRACTION   OF   LIGHT  IN  CALCITE 


89 


ii 


must  pass  through  //  wholly  as  an  ordinary  ray  (as  is,  accord- 
ing to  Fig.  40,  actually  the  case) ;  on  still  further  rotation,  again 
only  in  part  as  an  ordinary 
ray;  and  so  on.  In  order, 
therefore,  that  the  ray  e 
may  pass  through  //wholly 
as  an  ordinary  ray,  the 
plane  perpendicular  to  the 
principal  section  of  the 
second  calcite  must  be  par- 
allel to  the  principal  sec- 


Fig.  47- 


tion  of  the  first;  the  greater  the  angle  included  between  these  two 
planes,  the  smaller  is  the  part  of  the  ray  e  that  can  pass  through 
//  as  an  ordinary  ray.  So  the  plane  standing  perpendicular  to  the 
principal  section,  i.e.  the  transverse  plane  (see  p.  15)  of  the  ordi- 
nary ray,  is  a  character  of  the  extraordinary  ray  in  wholly  the 
same  way  as  the  principal  section  itself  is  a  character  of  the  ordi- 
nary; it  is  accordingly  the  polarization  plane  of  the  extraordinary 
ray:  this  ray  is  polarized  at  right  angles  to  the  principal  section. 
From  the  foregoing  it  is  seen  that  the  assumption  that  the 
ordinary  ray  vibrates  parallel  to  the  longer  diagonal  of  the 
rhombohedron  face  and  the  extraordinary  parallel  to  the  shorter, 
affords  a  perfect  explanation  of  the  observed  phenomena.* 

*  The  same  is  the  case  also  with  the  converse  assumption  —  that  the  extra- 
ordinary ray  carry  out  its  vibrations  parallel  to  the  longer  diagonal,  the  ordinary 
parallel  to  the  shorter.  For  if  in  Figs.  41  to  47  the  letters  o  and  e  are  substituted 
for  each  other  throughout,  exactly  the  same  deductions  result;  for  example,  from 
Fig.  41  or  Fig.  45  it  then  follows  that  the  ordinary  ray  emerging  from  the  first 
calcite  passes  wholly  as  an  ordinary  through  the  second,  and  so  on. 

These  two,  sole  possible  assumptions  differ  from  each  other  in  this:  that 
with  the  first  it  is  supposed  that  a  plane-polarized  light  ray  vibrate  perpendicular 
to  its  plane  of  polarization,  with  the  second  that  it  vibrate  parallel  to  that  plane. 
As  hitherto,  so  also  in  the  following,  it  will  always  be  assumed  that  the  vibration 
plane  of  a  plane-polarized  light  ray  stands  perpendicular  to  its  plane  of  polariza- 
tion. According  to  Maxwell's  electro-magnetic  theory  of  light  this  holds  true, 
properly  speaking,  only  of  the  electric  vibrations,  while  the  magnetic  take  place  in 
the  polarization  plane;  but  Wiener  has  demonstrated  that  for  the  optical  phe- 
nomena it  is  the  electric  vibrations  that  are  the  criterion. 


90  OPTICALLY   UNIAXIAL   CRYSTALS 

The  phenomena  in  question  are  further  consistent  with  the 
exposition  given  on  page  52,  according  to  which  the  resolution 
always  results  in  two  vibrations,  which  take  place  perpendicu- 
larly to  each  other.  But  this  exposition,  for  its  part,  is  based 
on  the  assumption,  made  on  page  16,  that  likewise  the  ordinary 
light  entering  the  calcite  is  plane-polarized  light  vibrating  per- 
pendicular to  the  direction  of  the  ray — light,  however,  whose 
vibration  plane,  or  what  signifies  the  same,  whose  plane  of 
polarization,  continuously  and  very  rapidly  changes.  That 
this  assumption  is  indeed  justified  is  proved  by  the  following 
experiment,  suggested  by  Dove.  When  a  calcite  rhombohedron 
upon  one  of  whose  faces  a  ray  of  ordinary  light  falls  at  right 
angles  —  in  consequence  whereof  the  arising  ordinary  ray  must 
emerge,  at  the  opposite  face,  along  the  same  straight  line  —  is 
caused  to  rotate  very  rapidly  about  this  line  as  an  axis,  the 
emerging  ordinary  ray  no  longer  exhibits  toward  a  second  cal- 
cite the  behavior  of  a  polarized  ray,  but  of  a  ray  of  ordinary 
light.  Here,  unquestionably,  we  have  to  do  with  a  plane- 
polarized  ray,  only  its  plane  of  polarization  (the  principal 
section  of  the  calcite)  is  caused  by  the  rotation  of  the  latter  to 
continuously  and  very  rapidly  change  its  direction. 

A  calcite  rhombohedron  exhibits  the  behavior  described  on 
page  Si  el  seq.,  whichever  of  its  three  pairs  of  faces  is  employed  as 
the  faces  of  entrance  and  emergence  of  the  light.  So  the  rhombo- 
hedron has  three  equivalent  principal  sections,  and  these  inter- 
sect one  another  along  the  direction  that  in  Fig.  48  is  placed 
vertical;  with  this  direction  the  three  upper,  and  consequently 
also  the  three  lower,  faces  of  the  rhombohedron  form  equal 
angles.  If  we  call  the  direction  A  A*  the  axis  of  the  rhombohe- 
dron, we  may  now  define  its  principal  optic  section  for  a  ray  fall- 
ing perpendicularly  on  a  rhombohedron  face  as  the  plane  that 
contains  the  axis  of  incidence  (i.e.  the  normal  to  the  rhombo- 
hedron face)  and  the  axis  of  the  rhombohedron.  Now  if  one 

*  If  we  think  of  it  as  a  line  connecting  two  opposite  corners  of  the  rhombohe- 
dron, then  at  each  of  its  extremities,  a,  there  intersect  three  equiangular  edges. 


DOUBLE   REFRACTION   OF   LIGHT   IN    CALCITE  9 1 

causes  light  rays  to  fall  in  such  a  principal  section,  not  perpen- 
dicular however  to  the  rhombohedron  face,  but  inclined  at  dif- 
ferent angles,  and  determines  for  each  ray  the  refractive  index 
of  the  two  rays  arising  by  double  refraction,  it  is  found  that  the 
index  of  the  ordinary  ray  (denoted  by  w) 
always  remains  the  same,  at  whatever 
angle  the  light  may  fall;  having,  namely, 
for  a  certain  medium  color  the  value 
1.6583.  This  was  indeed  to  be  expected, 
on  account  of  the  fact  that,  for  all  incli- 
nations of  the  incident  light  that  lie  in 
the  principal  section,  this  ray  has  always 
the  same  vibration  direction;  namely, 
according  to  our  assumption,  the  normal 
to  the  principal  section.  If  under  the 
same  conditions  one  determines  the  re- 
fractive index  of  the  extraordinary  ray,  * 
this  index  (e)  is  found  to  vary  with  the  FlS-  48- 
direction  in  which  the  ray  moves  in  the  calcite  (as  also  was  to  be 
expected,  since  the  vibration  direction,  falling  in  the  principal 
section  and  normal  to  the  transmission  direction,  varies  with  the 
inclination  of  the  incident  ray  to  the  axis).  This  refractive  index 
is  exactly  equal  to  that  of  the  ordinary  ray  (1.6583),  when  the 
extraordinary  is  transmitted  parallel  to  the  axis;  it  is  the  smaller, 
the  greater  the  angle  between  the  ray  direction  and  the  axis;  and 
has  its  least  value,  namely  1.4864,  when  the  ray  is  perpendicular 
to  the  axis. 

If  instead  of  one  of  the  three  above-defined  principal  sec- 
tions we  now  choose  any  plane  that,  like  those  three,  passes 
through  the  axis,  and  if  perpendicular  to  this  chosen  plane  we 
cut  on  the  calcite  a  plane  face  through  which  the  light  enters, 
then  we  observe  that  the  refractive  index  of  every  ray 
falling  at  whatever  inclination  to  the  axis  within  the  chosen 
plane  has  exactly  the  same  value  as  was  found  for  the  ray 
inclined  at  that  same  angle  within  a  principal  section.  Thus, 


92  OPTICALLY   UNIAXIAL   CRYSTALS 

for  this  plane  chosen  at  random  the  optical  relations  of  the 
crystal  are  wholly  the  same  as  for  the  three  above-defined 
principal  sections.  Consequently  this  random  plane  likewise 
may  be  spoken  of  as  a  principal  section;  and  hence  we  may 
designate  quite  generally,  as  the  principal  optic  section  for  an 
incident  ray,  that  plane  that  passes  through  the  ray  and  the  axis  of 
the  rhombohedron.  Calcite,  accordingly,  is  a  crystal  having  an  in- 
finity of  principal  optic  sections,  all  intersecting  one  another  in 
the  axis.  And  these  sections  are  all  equivalent;  for  in  each  of 
them  the  refractive  index  of  the  extraordinary  ray  diminishes  with 
increasing  inclination  to  the  axis  in  the  same  way,  while  that 
of  the  ordinary  retains  the  same  value  not  only  in  all  principal 
sections  but  also  in  all  directions  within  every  one  of  them. 

Since  a  smaller  refractive  index  corresponds  to  a  greater 
light  velocity  in  the  crystal,  there  follows  from  these  observations: 
THE  ORDINARY  RAY  o  arising  in  the  calcite  is  TRANSMITTED  EQUALLY 

FAST  IN  ALL  DIRECTIONS;  THE  EXTRAORDINARY  6  IN  THE  AXIAL 
DIRECTION  WITH  THIS  SAME  VELOCITY,  IN  EVERY  OTHER  DIRECTION 

WITH  A  GREATER  VELOCITY.  Since  the  two  rays  have  the  same  ve- 
locity when  they  move  parallel  to  the  axis,  then  parallel  to  this 
direction  there  can  (see  p.  51)  occur  no  double  refraction.  A 
direction  in  which  an  otherwise  doubly  refracting  crystal  be- 
haves like  a  singly  refracting  one,  i.e.  a  direction  in  which  its 
birefringence  is  zero,  is  called  an  optic  axis.*  Accordingly,  ordi- 
nary light  is  transmitted  through  the  calcite  as  ordinary  light 
when  it  goes  through  along  the  optic  axis;  i.e.  parallel  to  the 
axis  of  the  rhombohedron  (A A  in  Fig.  48). 

Quite  as  with  a  ray  of  ordinary  light  in  a  singly  refracting 
medium  is  the  behavior  of  the  ordinary  ray,  and  this  without 
exception;  so  the  direction  of  this  ray,  for  every  case,  is  to  be 
determined  by  Huygens's  construction:  ITS  RAY-SURFACE  is  A 

*  [It  is  to  be  borne  in  mind  that  an  "  optic  axis  "  is  nowise  a  line,  but  merely 
a  direction;  thus  too  with  a  "principal  optic  section":  it  is  not  a  definite  plane, 
but  any  one  of  an  infinity  of  parallel  planes.  The  same  applies  to  still  other  axes, 
sections,  etc.,  employed  in  crystal  optics;  for  it  is  obvious  that  in  a  crystal  all 
parallel  lines,  as  well  as  all  parallel  planes,  are  optically  equivalent.] 


DOUBLE    REFRACTION    OF    LIGHT   IN   CALCITE  93 

SPHERE.  As  for  the  extraordinary  ray,  Huygens  *  demonstrated 
that  the  different  velocities  of  the  extraordinary  rays  lying  within 
a  principal  section  but  differently  inclined  to  the  axis  are  to  one 
another  as  the  radii  vectores,  in  the  same  direction,  of  an  ellipse, 
whose  minor  axis  (i.e.  its  length)  is  the  velocity  of  the  extraordi- 
nary ray  parallel  to  the  optic  axis  and  whose  major  axis  is  the 
velocity  of  that  ray  perpendicular  to  the  axis.  Let  Y  (Fig.  49) 
be  a  direction  perpendicular  to  the  optic  axis,  X  a  direction 
parallel  to  that  axis,  the  plane  XY  therefore  a  principal  optic 
section  of  the  calcite;  and  let  further  ox  :  oy  be  the  ratio  be- 
tween the  light  velocities  of  the  ray  e  parallel  and  perpendicular 
to  the  optic  axis.  Then  the  velocity  o 
of  any  extraordinary  ray,  oR,  forming 
an  angle  <j>  with  the  optic  axis  is  or; 
i.e.  the  length  of  the  radius  vector  of 
the  ellipse  in  the  same  direction. 
Since  it  is  known  empirically  that  the 
light  velocity  of  the  extraordinary  ray 
is  the  same  for  all  directions  forming 
equal  angles  with  the  axis,  the  above  Flg*  49- 

applies  to  all  the  innumerable  principal  sections  that  may  be 
imagined  round  about  the  axis.  The  length  or  is  the  transmission 
velocity  for  all  rays  inclined  at  the  same  angle  (/>  — rays  obtained 
if  one  imagines  Fig.  49  to  be  rotated  360°  about  ox  as  an  axis. 
When  thus  rotated  the  whole  ellipse  yields  a  spheroid,  or  ellipsoid 
of  rotation,  whose  radius  vector  in  any  direction  is  a  measure  of 
the  velocity  of  e  in  that  direction.  If,  therefore,  from  any  point  o 
within  the  calcite  there  emanates  a  light-motion,  the  extraordi- 

*  Christian  Huygens:  ["Traite  de  la  lumiere.  Ou  sont  expliquees  les  causes 
de  ce  qui  lui  arrive  dans  la  reflection,  et  dans  la  refraction,  et  particulierement 
dans  1'etrange  refraction  du  cristal  d'Island."  Leyden,  1690.  A  recent  (1885) 
edition  is  still  in  print;  and  an  English  translation  of  the  first  three  chapters 
is  included  in  "The  Wave  Theory  of  Light",  edited  by  Henry  Crew.  New 
York,  1900.  (No.  X  of  the  Scientific  Memoirs  ed.  by  J.  S.  Ames.)  There  is  also 
a  German  edition  of  the  entire  treatise:]  "  Abhandlung  iiber  das  Licht ",  etc.,  ed.  by 
E.  Lommel.  Leipzig,  1890.  (Ostwald's  Klassiker  der  exacten  Wissenschaften 
Nr.  20.) 


94  OPTICALLY    UNIAXIAL    CRYSTALS 

nary  part  of  it,  although  it  will  be  transmitted  in  all  directions,, 
will  not  be  transmitted  in  all  directions  with  the  same  velocity: 
after  a  certain  time  it  will  have  been  transmitted  as  follows:  Along 
the  axis  as  far  as  x\  perpendicular  to  the  axis,  in  all  directions 
the  same  distance,  namely,  to  y\  along  the  random  direction  oR, 
as  far  as  r\  and  so  on  — in  a  word,  as  far  as  the  ellipsoid  sur- 
face generated  by  the  rotation  of  the  ellipse  xry  about  ox. 
Since  the  wave  length  of  light  increases  in  proportion  to  the 
transmission  velocity,  the  radii  of  this  ellipsoid  must  be  pro- 
portional to  the  several  wave  lengths  of  the  light  of  a  definite 
vibration  period,  the  light  of  which  several  wave  lengths  is 
transmitted  parallel  respectively  to  the  different  radii.  THE 

WAVE-  OR  RAY-SURFACE  OF  THE  EXTRAORDINARY  RAY  IS  ACCORD- 
INGLY A  ROTATION  ELLIPSOID,  WHOSE  MINOR  AXIS  IS  THE  TRANS- 
MISSION VELOCITY  OF  THE  RAY  PARALLEL  TO  THE  OPTIC  AXIS 

WHICH  IS  AT  THE  SAME  TIME  THE  ROTATION  AXIS  —  AND  WHOSE 
MAJOR  AXIS  IS  THE  VELOCITY  OF  6  PERPENDICULAR  TO  THE  OPTIC 

AXIS.  Since  in  the  axial  direction,  oX,  the  respective  transmis- 
sion velocities  of  the  ordinary  and 
the  extraordinary  ray  are  equal, 
then  in  the  direction  oX  the  ray- 
surface  of  the  former  must  have  the 
same  diameter,  ox,  as  that  of  the 
latter;  we  accordingly  obtain  the 
complete  ray-surface  of  the  light  in  i/ 
the  calcite  if  we  cause  the  two 

Fig.  50. 

curves  represented  in  Fig.  50   to 

rotate  about  the  vertical  axis.  If,  therefore,  a  light-motion  begins- 
at  any  point  within  the  calcite  and  can  advance  unhindered  in 
all  directions,  then  after  a  certain  time  it  has  arrived  at  a  double 
surface  which  consists  of  a  rotation'  ellipsoid  and  of  a  sphere 
enveloped  by  that  ellipsoid,  the  latter  touching  the  sphere  at 
the  extremities  of  the  rotation  axis  (the  minor  axis  of  the  gener- 
ating ellipse). 

By  our   knowledge  of   the   complete   wave-surface   we   are 


DOUBLE    REFRACTION    OF    LIGHT   IN    CALCITE 


95 


provided,  then,  in  Huygens's  construction,  with  a  means  of 
determining,  for  a  light  ray  entering  the  calcite  in  any  direc- 
tion, the  direction  of  the  extraordinary  ray  just  as  well  as  of  the 
ordinary. 

For  example:  If  abed  (Fig.  51)  be  a  principal  section  of  a 
calcite  rhombohedron  (ab  and  cd  short  diagonals,  ac  and  bd 
obtuse  edges),  then  is  the  direction  a  A  the  optic  axis;  hence 
if,  in  the  same  way  as  when  (see  Fig.  37)  for  the  first  time  we 
observed  the  phenomena  of 
double  refraction  in  calcite, 
there  fall  upon  ab  rays  of 
ordinary  light  with  perpen- 
dicular incidence,  their  ray- 
front  is  parallel  to  ab]  so 
they  will  all  begin  simulta- 
neously, starting  from  the 
points  m,  mv  m2,  .  .  .  re- 
spectively, to  be  transmitted 
in  the  calcite.  Proceeding 
from  these  points,  the  aris- 
ing ordinary  ray  arrives  in  a 
certain  time  at  the  surfaces 
of  the  spheres  o,  ov  02, .  .  .  ; 
according  to  a  previous 


Fig-  Si- 


page  its  new  ray-front  is  the  plane  tangent  in  common  to  all  these 
several  wave-surfaces,  namely  OO;  and  thus  the  (common)  direc- 
tion of  ma>,  m^aj^  etc.,  is  the  direction  of  the  arising  ordinary  rays. 
The  second,  the  extraordinary,  ray  has  in  the  same  time  advanced 
from  m,  ml}  mv  ...  to  the  surfaces  of  the  rotation  ellipsoids 
e,  ev  e2,  .  .  .,  whose  rotation  axis  is  parallel  to  aA\  conse- 
quently its  ray-front  is  the  tangential  plane  —  EE  in  our  sec- 
tion —  common  to  all  these  wave-surfaces;  so  the  direction, 
we,  m^v  etc.,  of  the  extraordinary  rays  is  given  by  the  straight 
lines  connecting  m,  mv  .  .  .  with  the  points  of  the  rotation 
ellipsoids  at  which  these  ellipsoids  are  touched  by  EE.  The 


96 


OPTICALLY   UNIAXIAL   CRYSTALS 


extraordinary  ray  must,  accordingly,  experience  a  deviation  in 
the  principal  section,  quite  as  we  have  previously  observed; 
and  one  sees,  besides,  that  with  this  ray  the  ray-front  and  the 
ray  direction  do  not,  as  with  the  ordinary  light  ray,  stand  per- 
pendicular to  each  other. 

This  is  due  to  the  circumstance  that  in  general  the  tangent 
to  an  ellipse  —  and  likewise  the  tangential  plane  tc-  a  spheroid 
—  does  not  stand  normal  to  the  radius  vector  at  whose  extremity 
it  touches  the  ellipse  or  the  ellipsoid.  The  obliqueness  of  the  two 

directions  (i.e.  radius  vector  and 
tangent),  and  thus  also  the  devi- 
ation of  the  extraordinary  ray,  is 
greatest  for  a  certain  inclination 
of  the  ray  to  the  axis,  this  incli- 
nation depending  on  the  form  of 
the  ellipse;  that  is,  on  its  axial 
ratio,  the  ratio  between  major 
and  minor  axis.  Only  at  the  ex- 
tremities of  these  two  axes  does 
the  tangent  stand  perpendicular 
to  the  radius  vector.  Therefore, 
if  the  face  of  entrance  ab  (Fig.  51) 
were  perpendicular  to  the  optic 
axis  of  the  calcite,  the-direction  of 


Fig.  52- 


the  light  rays  falling  normal  to  ab  would  coincide  with  Aa,  i.e. 
with  the  common  axis  of  both  skins  of  the  several  wave-surfaces; 
the  ray-fronts  tangent  to  these  skins  respectively  would  then  be 
identical,  as  would  likewise  the  directions  of  the  two  rays;  so  there 
would  then  be  only  one  ray-front  and  only  one  ray;  in  other  words, 
there  would  occur  no  double  refraction.  If  on  the  other  hand  the 
face  of  entrance  were  parallel  to  the  axis  (see  Fig.  52),  then  the 
two  ray-fronts,  OO  and  EE,  touching  the  several  wave-surfaces 
would  stand  at  their  greatest  distance  from  each  other,  this  corre- 
sponding to  the  greatest  difference  between  the  transmission 
velocity  of  the  ordinary  and  of  the  extraordinary  ray;  but,  since 


DOUBLE   REFRACTION   OF   LIGHT   IN   CALCITE 


97 


EE  touches  the  ellipses  at  the  end  of  an  axis,  then  me,  m^v  etc., 
are  perpendicular  to  EE;  consequently,  in  this  case  also,  the 
two  rays  arising  by  double  refraction  would,  in  the  crystal,  both 
have  the  same  transmission  direction.  Hence,  if  we  should 
imagine  plane-parallel  pairs  of  faces  of  various  inclinations  to  the 
axis  as  ground  on  a  piece  of  calcite  and  should  look  through  each 
pair  toward  a  small  bright  opening,  this  opening  would  appear 
single  when  we  used  the  pair  of  faces  perpendicular  to  the  axis 
(i.e.  when  the  rays  passed  through  the  calcite  parallel  to  the 


Fig-  53- 

axis);  when  the  pairs  of  faces  having  slight  inclination  to  the 
axis  were  used  two  images  would  appear,  whose  distance  apart 
would  increase  with  diminishing  obliqueness  of  the  faces  to  the 
axis,  reach  its  maximum  when  the  face  pair  through  which  the 
light  passed  formed  a  certain  angle  with  the  axis,  and  then 
diminish  again;  until  finally,  on  our  looking  through  two  faces 
parallel  to  the  axis,  the  two  images  would  be  seen  to  coincide. 

For  light  rays  falling  not  with  perpendicular,  but  with  oblique 
incidence  on  a  plane  face  of  the  calcite  the  simplest  case  is  that 
represented  in  Fig.  53,  where  the  principal  section  is  at  the  same 
time  the  plane  of  incidence;  in  other  words,  where  optic  axis, 


98  OPTICALLY   UNIAXIAL   CRYSTALS 

normal  to  the  face  of  entrance,  and  ray  direction  all  lie  in  one 
plane.  Here  the  points  w,  mlt  w2  of  the  face  of  entrance  are 
reached  by  the  wave-motion  at  different  instants;  the  front  of 
the  resulting  ordinary  ray  is  then  the  tangential  plane  through 
mO  perpendicular  to  the  principal  section,  and  that  of  the  ex- 
traordinary the  corresponding  plane  through  mE\  so  the  ordi- 
nary wave-motion  is  transmitted  along  the  direction  m^aj^  m2a)2, 
the  extraordinary  parallel  to  mlev  m2<-2.  The  ray-front  mE 
touches  the  several  ellipsoids  elt  e2  at  the  points  ep  £2;  these 
points  must  lie  in  the  plane  of  the  principal  section  abed,  since 
in  this  plane  there  falls  also  the  rotation  axis  of  the  ellipsoids 
and  since  this  principal  section,  which  is  at  the  same  time  the 
plane  of  incidence  of  the  light,  divides  each  of  the  ellipsoids  into 
two  symmetrical  halves. 

When  on  the  other  hand  the  plane  of  incidence  of  the  light 
does  not  coincide  with  a  principal  section,  it  no  longer  divides 
the  several  wave-surfaces  into  two  symmetrical  halves,  and  con- 
sequently the  tangential  plane  touches  those  surfaces  at  points 
that  lie  outside  the  plane  of  incidence.  The  extraordinary  ray, 
whose  direction  is  the  straight  line  connecting  the  center  of 
the  rotation  ellipsoid  with  the  point  at  which  the  ellipsoid 
is  touched  by  the  ray-front,  thus  leaves  the  plane  of  incidence 
in  every  case  where  this  plane  does  not  coincide  with  principal 
section;  the  ordinary,  of  course,  always  remaining  in  the  plane 
of  incidence,  since  its  wave-surface,  like  that  of  ordinary  light, 
is  a  sphere.  But  in  all  cases,  for  any  direction  of  the  incident 
light,  if  we  know  the  position  of  the  calcite  face  struck  by  the 
light  we  can,  from  the  position  and  form  of  the  wave-surface, 
determine  the  direction  not  only  of  the  arising  ordinary  ray  but 
also  of  the  extraordinary. 

Now  that  the  transmission  of  light  in  calcite  is  fully  under- 
stood, we  may  explain  the  contrivance  for  the  polarization  of 
light  mentioned  on  page  55,  the  so-called  Nicol  prism.*  This  is 
made  as  follows:  From  calcite  (cf.  footnote  I.e.)  a  parallelo- 

*  As  to  the  more  recent  improvements  in  the  Nicol  prism  cf.  footnote  f,  p.  30. 


DOUBLE    REFRACTION    OF   LIGHT   IN    CALCITE 


99 


piped  is  split  out  which  consists  of  four  larger  and  two  smaller 
faces  and  whose  principal  section  has  the  form  of  A  BCD  in  Fig. 
54.  After  the  upper  and  lower  end  faces  have  been  ground  down 
so  that  they  form  an  angle  of  68°  (instead  of  71°)  with  the  vertical 
edges,  the  parallelepiped  is  sawn  through  along  BC,  at  right 
angles  to  the  principal  section;  the  two  cut  surfaces  are  then 
made  perfectly  even  and  polished,  and  finally  the  two  halves 
cemented  together  again  in  their  original  position  with  Canada 
balsam.  A  light  ray  (r  in  Fig.  54)  falling  on 
such  a  prism  parallel  to  its  long  direction  is 
split  up  into  two  differently  refracted  rays.  The 
extraordinary  ray  e  moves,  in  the  calcite,  along 
a  direction  in  which  its  refractive  index  is  1.536; 
the  refractive  index  of  the  balsam  (since  this  is 
an  isotropic  body)  has  about  this  same  value  for 
all  kinds  of  vibrations,  and  consequently  the  ray  e 
will  pass  through  the  Canada  balsam  almost 
without  deviation.  The  ray  o,  on  the  other  hand, 
is  far  more  strongly  deflected,  impinging  therefore 
at  a  larger  angle  of  incidence  on  the  boundary 
of  calcite  and  Canada  balsam.  Now,  since  the 
refractive  index  of  this  ray  in  the  calcite  is  1.658 
and  in  the  balsam  1.536,  this  ray  is  transmitted 
in  the  first  medium  with  less  velocity  than  in  the 
second;  and,  according  to  page  34,  in  such  a  case 
the  light  can  pass  from  the  first  medium  into  the 
second  only  when  the  angle  of  incidence  does  not  exceed  a  cer- 
tain limit.  In  the  case  described  this  limit  is  exceeded,  in  con- 
sequence whereof  the  ray  o  is  totally  reflected;  thereby  it  is 
thrown  upon  the  lateral  faces  of  the  prism  and  absorbed  by  the 
blackened  mounting  of  the  same.  Entirely  through  the  nicol, 
therefore,  there  passes  only  the  extraordinary  ray,  vibrating  in  the 
plane  of  the  principal  section  and  polarized  perpendicular  to  it. 
On  page  91  it  was  stated  that  in  calcite,  for  the  light  of  a 
certain  medium  color,  the  refractive  index  a>  of  the  ordinary  ray 


Fig.  54- 


IOO  OPTICALLY  UNIAXIAL  CRYSTALS 

has  the  constant  value  1.6583;  and  that  the  index  e  of  the  ex- 
traordinary ray,  when  the  transmission  direction  is  perpendic- 
ular to  the  axis,  has  that  one  of  its  values  that  differs  the  most 
from  a)j  — namely,  1.4864.  These  two  quantities  are  called  the 
principal  refractive  indices  of  calcite  for  the  color  in  question. 
If  we  denote  by  v  the  transmission  velocity  of  the  light  of  this 
color  in  air,  by  v0  that  of  the  ordinary  ray  of  the  same  vibra- 
tion period,  and  finally  by  ve  the  velocity  of  the  extraordinary  light 
ray  for  the  given  crystallographic  direction  and  the  same  color, 
then  according  to  the  general  definition  of  the  refractive  index 
(see  p.  33)  we  have 

v  v 

a)  =  1.6583  =  —  >  e  =  1.4864  =  —  • 

V0  Ve 

Thence  follows  the  ratio  between  the  two  principal  transmis- 
sion velocities: 

*-6$Sl 

v-:v--w:f"7^64         "S*- 

This  ratio  ve  :  v0,  according  to  page  93,  is  at  the  same  time  the 
ratio  of  the  major  to  the  minor  axis  of  the  ellipse  generating  the 
wave-surface  of  the  extraordinary  ray;  and  the  above  value  of 
it  applies  to  the  line  D  in  the  spectrum,  since  the  numerical 
values  mentioned  for  w  and  s  refer  to  this  line.  Light  rays  of 
a  different  vibration  period  yield  a  somewhat  different  axial 
ratio  of  that  generating  ellipse.  If  we  determine  the  values  of 
the  two  principal  refractive  indices  in  calcite  for  light  of  less 
refrangibility,  e.g.  for  the  line  A  in  the  red,  we  find  aj  =  1.6499, 
e  =  1.4826;  and  therefore  the  ratio 

1.6499  o 

ve  :v0  =  a)  : e  =  — ;—-  =  1.1128. 
1.4826 

This  value  is  somewhat  smaller  than  the  last.  Conversely,  for 
light  of  greater  refrangibility  a  higher  value  results;  e.g.  for  the 
line  H  in  the  violet  a>  =  1.6832  and  e  =  1.4977,  s°  tnat  tne  rati° 

1.6832 
ve  :  V0  =  co  :  e  = J—  =  1.1238. 

1.4977 


DOUBLE   REFRACTION  IN   OTHER   UNI  AXIAL   CRYSTALS      IOI 

Hence  it  follows  that  the  form  of  the  ray-surface  varies  with 
the  color  of  the  light,  and  that  the  double  refraction  becomes 
stronger  with  diminishing  wave  length.  :  I  -TV  *  v  u 

It  is  customary  to  designate  as  strength  of-  double  .refr^ie*-. 
tion,  or  as  birefringence,  the  difference  between  the  two  principal 
refractive  indices;  this  difference  has  in  calcite,  for  the  three  colors 
named,  the  following  values: 

Line  A  :  to  —  e  =  0.1673 
Line  B  :  to  —  e  =  0.1719 
Line  C  :  QJ  —  £  =  0.1855 

DOUBLE  REFRACTION  OF  LIGHT  IN  OTHER  UNIAXIAL 
CRYSTALS 

The  fact  established  first  for  calcite,  by  Huygens  (although  at 
that  time  — in  the  year  1678  — not  with  such  exact  numerical 
values  as  those  given  above),  the  fact,  namely,  that  the  ray- 
surface  of  light  in  the  crystal  consists  of  a  sphere  and  of  a  rota- 
tion ellipsoid,  the  latter  touching  the  former  at  the  extremities  of 
its  rotation  axis,  has  later  been  demonstrated  for  numerous  other 
crystals.  These  all  have  in  common  with  calcite  that  the  form 
of  the  ray-surface  depends  on  the  vibration  period  of  the  light; 
but  for  each  of  these  kinds  of  crystals  the  manner  in  which  the 
form  of  the  ray-surface  varies  with  the  color  is  different,  just  as 
also  the  values  of  the  two  principal  refractive  indices,  at  and  e, 
are  different  for  the  same  color  with  different  crystallized  sub- 
stances of  this  kind.  Thus,  among  them  are  crystals  in  which 
co  and  s  have  higher  values  than  in  calcite;  others,  with  lower 
refringency;  further,  crystals  with  higher  (among  these  belongs 
calcite)  and  with  lower  birefringence,  in  which  latter  there- 
fore to  and  s  differ  but  little.  (E.g.  in  penninite  co-  e  =0.001.) 

But  in  each  of  these  crystals,  as  in  calcite,  the  direction  of 
the  rotation  axis  of  the  spheroid  of  the  ray-surface,  i.e.  the  direc- 
tion without  double  refraction,  is  the  same  for  all  colors.  These 
crystals,  therefore,  have  only  one  optic  axis,  and  for  that  reason 
are  designated  as  optically  uniazial  crystals. 


102 


OPTICALLY   UNIAXIAL   CRYSTALS 


In  one  respect,  however,  do  only  a  part  of   the  optically 

uniaxial, crystals  agree  with  calcite;   namely,  in  that  with  them 

**  *     * .  *»  ?   * 


the  transmission  velocity  of  the  extraordinary  light  ray  is  greater 
than  that  of  the  ordinary,  while  with  the  rest  the  contrary  is  the 

case.  The  optically  uniaxial  crys- 
tals accordingly  fall  into  two  divi- 
sions. 

The  crystals  of  the  first  division, 
among  which  calcite  belongs,  are 
those  whose  wave-surface,  bisected 
along  the  principal  section,  has  the 
form  shown  in  Fig.  50  on  page  94, — • 
except  that  with  the  majority  of 
them  the  ellipse  diverges  far  less 
from  the  circle  than  is  the  case 
with  calcite.*  As  is  manifest  from 

Fig.    55,    in    these    crystals    the    extraordinary    ray   e    is,    as 
reckoned  from  the  ordinary  0,  always  refracted  away  from  the 


*  With  calcite,  also,  the  difference  between  major  and  minor  axis  of  the  ellipse  is 
not  so  considerable  as  it  has,  for  the  sake  of  distinctness,  been  assumed  in  the  figure. 


DOUBLE   REFRACTION   IN   OTHER   UNIAXIAL   CRYSTALS      103 

optic  axis  A  A  ;  they  are  for  this  reason  called  repulsive,  or 
negative  crystals. 

In  a  crystal  of  the  second  division,  where,  for  example,  quartz 
belongs,  the  wave-surface  has,  in  the  principal  section,  the  form 
shown  in  Fig.  56  on  the  opposite  page:  in  the  axial  direction, 
only,  the  transmission  velocity  of  the  extraordinary  ray  is  equal 
to  that  of  the  ordinary;  in  every  other  direction  it  is  less,  and 
the  refractive  index  therefore  greater,  than  that  of  the  ordinary. 
From  Fig.  57,  then,  may  be  gathered  that  here,  contrary  to 


what  is  the  case  with  the  negative  crystals,  the  extraordinary 
ray  e,  as  reckoned  from  the  ordinary,  is  refracted  toward  the 
optic  axis.  These  crystals  are  therefore  designated  as  optically 
positive,  or  attractive. 

To  whichever  of  the  two  divisions  an  optically  uniaxial 
crystal  may  belong,  from  its  ray-surface  the  transmission  direc- 
tion of  any  ray  can  be  found  by  Huygens's  construction,  in  ex- 
actly the  same  way  as  was  shown  for  calcite  on  page  95  et  seq. 
With  regard  also  to  the  vibration  directions  of  the  two  rays  aris- 
ing by  the  double  refraction,  all  uniaxial  crystals  conform  to  the 
same  laws:  in  them  a  vibration  taking  place  in  any  direction  is 


104  OPTICALLY   UNI  AXIAL   CRYSTALS 

always  resolved  into  two;  viz.  (i)  an  ordinary  vibration,  polarized 
in  the  plane  of  the  principal  section  and  therefore,  according  to 
our  previous  assumption,  directed  perpendicular  to  it,  and  (2)  an 
extraordinary  vibration,  whose  polarization  plane  is  the  plane 
that  passes  through  the  ray  perpendicular  to  the  principal 
section, —  this  latter  vibration,  therefore,  according  to  the  same 
assumption,  taking  place  in  the  plane  of  the  principal  section 
and  perpendicular  to  the  ray.  Since  the  principal  section  is 
always  denned  by  the  ray  direction  and  the  optic  axis,  but  since 
both  these  directions  follow  from  the  form  of  the  ray-surface  and 
its  crystallographic  orientation,  this  surface  supplies  us  also  with 
a  knowledge  of  the  vibration  direction,  or  say  the  polarization, 
of  any  light  ray. 

In  order  to  determine  the  form  of  the  ray-surface,  it  is  neces- 
sary to  measure  the  two  velocities  with  which  the  extraordinary 
ray  is  transmitted  when  it  passes  through  the  crystal  once  paral- 
lel, once  perpendicular r  to  the  axis;  for  the  ratio  between  these 
two  magnitudes  is  the  axial  ratio  of  the  ellipse  generating  the 
ray-surface,  by  which  ratio  the  form  of  the  ray-surface  is  com- 
pletely determined.  But,  since  the  former  velocity  is  identical 
with  that  of  the  ordinary  ray,  this  ratio  is  v0  :  ve;  i.e.  the  inverse 
ratio  of  the  two  principal  refractive  indices  a>  and  e.  (Cf.  p.  100.) 
Hence  one  needs  only  to  measure  the  refractive  index  of  the 
ordinary  ray  for  any  direction,  and  that  of  the  extraordinary 
when  it  is  transmitted  perpendicular  to  the  axis;  i.e.,  according 
to  our  assumption,  the  refractive  index  CD  of  light-vibrations  that 
take  place  perpendicular  to  the  axis,  and  the  index  e  of  rays 
vibrating  parallel  to  the  axis.  This  may  be  done  in  various 
ways:  - 

i.  By  means  of  a  prism  whose  refracting  edge  is  parallel  to 
the  optic  axis.  In  such  a  prism,  when  the  method  set  forth  on 
page  39  is  employed,  the  plane  of  incidence  of  the  light  stands  per- 
pendicular to  the  optic  axis;  therefore  an  incident  ray  is  resolved 
into  two  rays  of  which  the  one  vibrates  parallel  to  the  optic  axis, 
the  other  perpendicular  to  this  axis;  substituted  in  the  formula 


DOUBLE  REFRACTION  IN  OTHER  UNIAXIAL   CRYSTALS      105 

given  /.  c.,  the  minimum  deviation  of  the  former  ray  supplies  e, 
that  of  the  latter  ray  a>.  The  two  rays  can  easily  be  distin- 
guished from  each  other  with  the  aid  of  a  Nicol  prism  held 
before  the  eye;  for  when  the  principal  section  of  the  nicol  is 
parallel  to  the  refracting  edge  of  the  prism,  the  extraordi- 
nary ray  only  is  transmitted,  the  ordinary  on  the  other  hand 
extinguished. 

2.  By  means  of  a  prism  whose  refracting  edge  is  perpendic- 
ular to  the  optic  axis.     But  here  this  further  condition  must  be 
fulfilled:    that  the  two  faces  of  the  prism  include  equal  angles 
with  the  axis,  in  order  that  the  rays  transmitted  in  the  prism 
may  in  their  minimum  deviation  be  directed  normal  to  the  optic 
axis.     (Cf.  Fig.  20,  p.  38,  in  which,  applied  to  this  case,  Mm 
would  be  parallel  to  the  optic  axis.)     For  then  the  vibration 
direction  of  the  one  ray  is  again  parallel,  that  of  the  other  again 
perpendicular,  to  the  axis.     With  such  a  prism  one  can  de- 
termine e  and  aj  at  the  same  time,  by  adjusting  each  of  the  two 
rays,  singly,  at  its  minimum  deviation  and  calculating  the  re- 
fractive index  proper  to   that  ray  from  the  formula  given  on 
page  38. 

3.  By  means  of  the  total-reflectometer  (see  p.  40  et  seq.)  and 
a  plate  cut  from  the  crystal  in  any  direction.     Herewith  one 
obtains  two  boundaries  of  total  reflection,  because  the  ray  trans- 
mitted in  the  boundary-layer  of  the  plate  is  resolved  into  two  rays 
which  vibrate  perpendicularly  to  each  other  and  which  have,  in 
correspondence  to  the  difference  in  their  velocity,  different  critical 
angles   of  total  reflection.     Whatever    be    the   crystallographic 
orientation   of   the    plate,  there    always    exists    in    its   plane  a 
direction    forming  90°  with  the  optic    axis;    hence,    if  one   so 
fastens  the  plate  in  the  total-reflectometer  that  this  direction  is 
horizontal,  i.e.  parallel  to  the  plane  of  incidence  of  the  light,  it 
is  along  this  direction  that  the  light  is  transmitted  when  total 
reflection  occurs.     But  a  ray  transmitted  in  the  crystal  perpen- 
dicular  to   the  axis  is  resolved   into  two   rays,  of  which  one 
vibrates  parallel,  the  other  perpendicular,  to  the  axis.     There- 


106  OPTICALLY   UNIAXIAL   CRYSTALS 

fore  the  adjusting  of  the  two  boundaries  of  total  reflection 
supplies  w  and  e. 

If  the  determination  of  the  two  principal  refractive  indices 
is  carried  out  by  one  of  these  methods  for  two  homogeneous 
colors,  as  lithium-red  and  sodium-yellow,  then  from  the  values 
obtained  we  may  calculate  the  refractive  indices  for  all  the  re- 
maining colors,  according  to  the  dispersion  formula  given  on 
page  46.  But,  since  the  two  rays  vibrating  perpendicularly  to 
each  other  are  differently  refracted  and  also,  as  is  learned  em- 
pirically, always  unequally  dispersed,  the  constants  A  and  B 
of  the  dispersion  formula  are,  of  course,  not  the  same  for  a>  as 
for  e. 

Besides  the  determination  of  the  form  of  the  ray-surface  for  one 
or  more  colors  there  is  requisite,  for  the  complete  optical  knowl- 
edge of  a  crystal,  that  the  orientation  of  its  optic  axis  be  ascer- 
tained. The  quest  of  this  direction  takes  place  with  the  aid  of 
the  interference  phenomena  in  convergent  polarized  light,  where- 
fore these  phenomena  shall  be  treated  in  the  following  section, 
in  conjunction  with  those  observed  in  parallel  light. 

BEHAVIOR  OF    UNIAXIAL  CRYSTALS  IN  THE    POLARIZATION 

APPARATUS 

a.  IN  PARALLEL  LIGHT  (ORTHOSCOPE)  . —  If  into  horizontal 
position  upon  the  stage  of  the  orthoscope  (see  Fig.  34,  p.  75)  we 
bring  a  crystal  plate  bounded  by  two  natural  or  artificial  plane 
surfaces  that  are  parallel  to  each  other  and  perpendicular  to  the 
optic  axis,  then  through  the  plate  there  pass  only  such  rays  as 
are  vertical  in  direction  and  therefore  parallel  to  the  optic  axis. 
But  this  is  the  direction  in  which  there  is  no  double  refraction, — 
the  direction  in  which  a  uniaxial  crystal  behaves  like  a  singly 
refracting  body.  Accordingly  the  plate  does  not  in  the  least 
degree  alter  the  polarization  of  the  rays  passing  through  it:  it 
appears  just  as  dark  as  the  rest  of  the  field  when  the  nicols  are 
crossed  at  right  angles,  just  as  bright  with  the  nicols  parallel. 
But  in  the  first  case  the  plate  naturally  appears  absolutely  dark 


BEHAVIOR   IN   THE   POLARIZATION   APPARATUS  107 

only  when  no  rays  pass  through  other  than  such  as  are  exactly 
parallel  to  the  optic  axis.  This,  strictly  speaking,  is  the  case 
only  with  microscopic  crystals,  because  through  larger  plates 
rays  still  find  their  way  that  are  more  or  less  inclined  to  the  axis; 
these  rays  experience  double  refraction  and  consequently  a 
brightening. 

When  the  faces  of  incidence  and  emergence  are  not  perpen- 
dicular to  the  optic  axis,  double  refraction  takes  place  also  for 
rays  that  fall  normal  to  the  plate,  since  these  rays  now  form 
with  the  optic  axis  an  angle  other  than  zero.  Every  incident 
ray  is  resolved  on  the  one  hand  into  an  ordinary  ray,  vibrating 
at  right  angles  to  the  principal  section  and  passing  through 
without  deviation;  and  on  the  other  hand  into  an  extraordinary 
ray,  which,  vibrating  in  the  principal  section  and  deviated  in 
this  same  plane,  is,  on  emerging  from  the  plane-parallel  plate, 
again  refracted  so  that  it  becomes  once  more  parallel  to  its  pre- 
vious direction,  • — i.e.  to  the  normal  to  the  plane  of  the  plate. 
If  we  imagine  the  whole  crystal  plate  as  illuminated  by  rays 
falling  perpendicular  to  it,  then  from  every  point  of  the  face  of 
emergence  there  will  be  transmitted  normal  to  the  plate  two 
light  rays;  namely,  an  ordinary,  which  has  passed  through  .the 
plate  at  right  angles,  and  an  extraordinary,  which  within  the 
crystal  moved  oblique  to  the  faces  of  entrance  and  emergence 
and  hence  must  have  been  derived  from  another  one  of  the  inci- 
dent rays  than  was  the  ordinary  emerging  at  that  same  point. 
The  two  rays,  transmitted  in  the  air  along  the  same  path,  vibrate 
perpendicularly  to  each  other  and,  because  of  their  difference  in 
velocity  within  the  crystal,  have  a  difference  of  path;  so  with 
crossed  nicols  they  must  exhibit  the  interference  phenomena 
treated  in  detail  on  pages  64-70.  The  strength  of  the  double  re- 
fraction in  the  crystal  plate  depends  on  the  inclination  of  the  nor- 
mal of  the  plate  to  the  optic  axis :  when  that  inclination  is  small, 
so  also  is  the  double  refraction,  and  the  colors  of  low  order 
appear;  with  equal  plate  thickness  the  order  of  the  color  rises  with 
increasing  amount  of  that  inclination,  and  is  highest  in  the  case  of 


108  OPTICALLY   UNIAXIAL   CRYSTALS 

a  plate  parallel  to  the  axis;  for  in  such  a  plate  the  two  rays  are 
transmitted  perpendicular  to  the  axis  and  therefore  acquire  the 
greatest  difference  of  path.*  If  we  investigate  plates  of  differ- 
ent thickness,  the  path  difference  naturally  increases,  and  with  it 
the  order  of  the  color,  — proportionally  to  the  thickness  if  the 
plates  have  equal  inclination  to  the  optic  axis, —  until  at  last  the 
white  of  a  higher  order  appears.  But  the  plate  thickness  with 
which  this  occurs  is  obviously  the  greater,  the  smaller  the  angle 
formed  by  the  light  rays  with  the  optic  axis,  i.e.  the  less  the 
double  refraction;  therefore,  of  all  the  plates  that  can  be 
imagined  as  cut  from  a  crystal  in  different  directions,  those  cut 
parallel  to  the  axis  exhibit  the  white  of  a  higher  order  with 
relatively  the  least  thickness. 

If  we  compare  crystal  plates  of  different  substances  with 
one  another,  then,  even  with  equal  thickness  and  equal  inclina- 
tion to  the  optic  axis,  these  plates  exhibit  a  different  inter- 
ference-color; for  naturally  the  color  exhibited  depends  more- 
over on  the  birefringence,  or  specific  double  refraction,  of  each 
substance,  — i.e.  on  the  relative  distance  of  the  two  skins  of  its 
ray-surface  from  the  center.  When  this  distance  is  slight,  there 
is  requisite  either  a  great  thickness  of  the  crystal  plate  or  else  a 
considerable  inclination  of  its  normal  to  the  optic  axis,  in  order 
to  produce  the  white  of  a  higher  order;  on  the  other  hand,  a 
crystal  of  high  birefringence  exhibits  this  white,  even  with 
slight  thickness  or  when  there  is  only  a  small  angle  between  the 
axis  and  the  normal  to  the  plate. 

•*.  Finally,  in  addition  to  depending  on  the  conditions  enumer- 
ated, the  color  phenomena  that  a  uniaxial  crystal  presents  in 
polarized  light  are  influenced  by  the  circumstance  that  the 

*  Therefore,  if  in  a  very  thin  plane-parallel  plate  made  from  a  rock  —  a 
so-called  "thin  section"  of  the  rock  —  there  lie  sections  of  a  number  of  diversely 
oriented  crystals  of  an  optically  uniaxial  mineral,  these  crystal  sections,  in  spite  of 
their  having  equal  thickness,  must  according  to  the  above  exhibit  different  inter- 
ference-colors; and  those  crystals  that  the  cut  happens  to  have  intersected  parallel 
to  the  axis  will  exhibit  the  highest  in  order  of  the  interference-colors  described 
on  p.  64  et  seq. 


BEHAVIOR   IN   THE   POLARIZATION    APPARATUS  109 

birefringence  is  not  exactly  equal  for  the  different  colors  (i.e. 
that  the  ordinary  and  extraordinary  rays  of  white  light  experi- 
ence unequal  dispersion),  thin  plates  of  the  crystal  therefore 
exhibiting  smaller  or  larger  deviations  from  the  normal  inter- 
ference-colors. (Cf.  p.  69.)  This  circumstance  becomes  of 
importance  in  cases  where  the  birefringence  is  very  low;  and 
here  it  can  even  lead  to  phenomena  quite  different  from  those 
otherwise  observed.  For  example,  if  with  increase  in  the  re- 
frangibility  of  the  rays  the  birefringence  become  greater,  but 
still  be  so  slight  that  for  the  least  refrangible  colors  —  thus  for 
red  — it  is  zero,  then  a  very  thin  plate  of  such  a  crystal  will  not 
exhibit  the  gray  of  the  first  order,  corresponding  to  the  very 
slight  birefringence  proper  to  it,  but  a  very  vivid  color;  so  that 
from  the  appearance  it  looks  as  though  the  substance  in  ques- 
tion had  very  strong  double  refraction.  The  explanation  is  as 
follows:  Since  the  crystal  plate  is  singly  refracting  for  red,  it 
remains  for  this  color,  between  crossed  nicols,  dark  in  every 
position,  the  red  rays  being  totally  annihilated;  further,  for 
orange  and  yellow  the  plate  exhibits  only  a  slight  brightening; 
somewhat  more  light  passes  through  from  the  green,  while  for 
blue  and  violet  the  difference  of  path  is  so  great  that  even  after 
the  interference  these  colors  still  have  a  certain  intensity;  the 
latter  colors  must  therefore  greatly  predominate  in  the  color 
impression.  The  arising  interference-color  must  be  still  more 
vivid  when  the  birefringence  is  so  low  that  even  for  a  medium 
color,  e.g.  yellow,  it  is  zero;  the  crystal  accordingly  then  has,  for 
the  colors  lying  at  opposite  ends  of  the  spectrum,  double  refrac- 
tion of  opposite  character:  for  the  colors  of  one  end  it  is  posi- 
tively uniaxial,  for  those  of  the  other  end  negatively  uniaxial. 
Hence,  if  we  bring  a  thin  plate  of  such  a  crystal  between  crossed 
nicols  and  turn  it  so  that  its  vibration  directions  form  45°  with 
those  of  the  nicols,  the  yellow  rays,  the  brightest  rays  of  the 
spectrum,  are  totally  extinguished,  because  for  them  the  crystal 
is  singly  refracting;  for  the  adjacent  colors,  orange  and  green, 
it  is  but  feebly  birefringent  and  so  for  these  exhibits  only  a 


no 


OPTICALLY   UNIAXIAL  CRYSTALS 


slight  brightening;  while  for  the  outermost  parts  of  the  spectrum 
—  red  on  the  one  hand,  blue  on  the  other  — the  double  refrac- 
tion is  relatively  strongest,  and  the  rays  of  these  colors  accord- 
ingly acquire  a  path  difference  amounting  to  a  perceptible 
fraction  of  a  half  wave  length.  The  consequence  is  that  be- 
tween crossed  nicols  a  very  vivid  violet  interference-color  ap- 
pears. (An  excellent  example  of  this  phenomenon  is  exhibited 
by  certain  varieties  of  the  mineral  gehlenite.) 

b.  IN  CONVERGENT  LIGHT  (CONOSCOPE)  .  —  As  in  parallel 
polarized  light,  so  also  in  convergent,  the  behavior  of  a 
uniaxial  crystal  will  be  considered  first  for  a  plate  cut  per- 

f 


Fig-  58. 

pendicular  to  the  optic  axis.  If  we  lay  such  a  crystal  plate  upon 
the  stage  of  the  conoscope  (cf.  Fig.  35,  p.  78)  in  such  a  way 
that  the  optic  axis  is  parallel  to  the  vertical  axis  of  the 
whole  apparatus,  and  that  the  point  /  lies  within  the  crystal, 
then  an  infinity  of  ray  bundles,  each  bundle  consisting  of 
mutually  parallel  rays,  pass  through  the  crystal  in  all  pos- 
sible directions  within  the  cone  between  /  and  the  so-called 
condensing  lens  n.  One  of  these  ray  cylinders  —  namely,  the 
one  proceeding  from  the  point  c  and  therefore  appearing  in 
the  image  exactly  at  the  center  of  the  field  —  passes  through  the 


BEHAVIOR  IN  THE   POLARIZATION  APPARATUS,          III 

crystal  parallel  to  its  optic  axis.  In  order  to  trace  the  changes 
experienced  within  the  plate  by  all  these  differently  directed  ray 
bundles,  let  us  consider  first  the  rays  that" lie  in  one  vertical  plane 
passing  through  the  optic  axis  of  the  crystal.  Such  a  plane  we 
called  a  principal  section  of  the  uniaxial  crystal,  and  in  Fig.  58 
this  shall  be  represented  by  MNOQ.  Let  us  then  suppose,  in 
addition,  that  the  two  nicols  of  the  instrument  are  crossed,  that 
their  polarization  planes  form  45°  with  the  principal  section 
MNOQ,  and  that  monochromatic  light  is  passing  through  the 
apparatus.  The  light  rays  parallel  to  the  optic  axis  of  the  crystal, 
the  direction  AB,  are  not  doubly  refracted,  but  pass  through  the 
crystal  unaltered ;  so  the  central  part  of  the  image  appears  dark, 
exactly  as  would  the  whole  field,  with  crossed  nicols,  if  no  crystal 
plate  were  present.  But  if  we  now  consider  the  ray  cylinders 
that  have  a  slight  inclination  to  the  optic  axis,  then  among  these 
cylinders  there  will  be  one  —  the  one  to  which,  for  example, 
the  ray  CD  belongs — to  which  the  following  applies:  The  ray 
CD  is  split  up  in  the  crystal  into  two  rays,  DH  and  DJ,  of 
different  velocity,  one  of  these  rays  vibrating  in  the  principal 
section  MNOQ,  the  other  perpendicular  to  it;  just  so  does 
another  ray  EF,  of  the  same  bundle  and  therefore  parallel  to 
CD,  split  up  into  an  ordinary  and  an  extraordinary  ray,  FG  and 
FH.  Thus,  from  H  there  proceed  farther,  along  the  same 
path,  two  rays  which  both  came  from  the  same  source  of  light, 
a  source  emitting  plane-polarized  light, —  namely,  from  the  point 
in  question  of  the  bright  opening  de  in  Fig.  35,  —  but  which  are 
polarized  perpendicularly  to  each  other.  Of  each  of  these 
vibrations,  only  that  component  is  transmitted  through  the 
upper  nicol,  that  falls  to  its  vibration  plane;  so  we  have  the  case 
represented  in  Fig.  32,  page  61.  And  if  the  inclination  of  the 
considered  rays  (CD  and  EF)  to  the  optic  axis  of  the  crystal  is 
such  that  the  path  difference  of  the  arising  light  rays  DH  and 
FH  amounts  to  exactly  a  half  wave  length  of  the  color  with 
which  the  instrument  is  illuminated,  then,  according  to  page  60, 
these  rays  DH  and  FH  will  combine  by  the  interference  to 


112 


OPTICALLY   UNIAXIAL   CRYSTALS 


form  the  sum  of  their  transmitted  components.  Now  among  the 
parallel  rays  that  traverse  the  crystal  in  the  same  direction  as 
the  two  just  considered,  CD  and  EF,  there  must  exist  for  every 
ray  of  them  a  ray  that  stands  to  this  first  ray  in  the  same  rela- 
tion as  the  relation  between  CD  and  EF;  that  is,  the  ordinary 
ray  arising  from  the  one  interferes  with  the  extraordinary  aris- 
ing from  the  other  in  the  same  way  as  do  DH  and  FH.  Hence, 
in  the  image  visible  in  the  conoscope,  at  the  point  where  all 


these  rays  converge — a  point  somewhat  removed  from  the  center 
of  the  field  —  there  will  appear  the  brightness  corresponding  to 
the  sum  of  the  transmitted  components  of  all  these  pairs  of 
interfering  rays.  If  Fig.  59  represent  the  image  seen  in  the 
polarization  apparatus,  and  if  A  A'  and  PP'  be  the  vibration 
directions  of  the  two  crossed  nicols,  MN  the  direction  of  the 
principal  section  MNOQ  of  the  last  figure,  then  is  C  the  dark 
center  of  the  image  and  bl  the  bright  place  at  which  those  rays 
converge  that  were  last  discussed.  The  positions  between  C 
and  6j  represent  the  points  of  convergence  for  ray  cylinders  that 


BEHAVIOR   IN   THE   POLARIZATION   APPARATUS  113 

have  a  smaller  inclination  to  the  optic  axis,  so  that  for  these 
the  path  difference  acquired  in  the  crystal  is  less  than  \X. 
Such  rays,  on  the  interference,  will  combine  to  form  a  wave- 
motion  whose  state  of  vibration  is  different  from  that  of  either 
component,  and  whose  intensity  must  accordingly  be  less  than  the 
sum  of  the  intensities  of  the  single  rays  (see  p.  19) ;  the  result- 
ing intensity  approaches  the  more  closely  to  this  latter  value, 
however,  the  closer  we  come  to  bv  —  that  is,  the  less  the  differ- 
ence of  path  differs  from  J  L  So  from  bl  to  the  center,  where  it 
is  zero,  the  brightness  must  gradually  diminish.  With  rays  in 
the  principal  section  MO  that  are  inclined  to  the  optic  axis  of  the 
crystal  at  a  greater  angle  than  the  rays  converging  at  b19  the 
difference  in  the  transmission  velocity  of  the  two  arising  vibra- 
tions is  greater  than  with  the  latter  rays,  since  the  velocity  of 
the  extraordinary  ray  differs  the  more  from  that  of  the  ordinary, 
the  more  the  ray  is  inclined  to  the  axis.  Therefore  at  a  point  dlt 
on  the  straight  line  MN  but  farther  removed  from  the  center, 
there  will  converge  all  rays  whose  direction  in  the  crystal  was 
such  that  the  ordinary  and  the  extraordinary  arising  from  each 
acquired  a  path  difference  of  L  But,  according  to  page  57,  on 
the  interference  between  crossed  nicols  each  of  these  two  arising 
vibrations  and  the  vibration  with  which  it  interferes,  which 
arose  from  a  different  ray  of  the  same  direction,  —  thus  all  the 
rays,  pairwise,  that  arise  from  rays  of  this  direction,  —  must  com- 
pletely annihilate  each  other.  Therefore  the  point  dl  will  appear 
just  as  dark  as  does  the  center  of  the  image;  and  the  bright- 
ness will  gradually  diminish  from  b1  to  d^  but  on  the  other 
side  of  d^  it  will  increase,  since  the  point  62  corresponds  to  the 
convergence  of  those  rays  that  passed  through  the  crystal  so 
inclined  that  the  path  difference  of  the  two  rays  arising  by  the 
double  refraction  is  f  ^,  and  that  accordingly  on  the  interference 
a  superposing  of  the  two  components  again  takes  place.  The 
same  is  the  case  at  &3,  where  the  path  difference  is  §L  And  so 
on.  If,  therefore,  proceeding  outward  from  the  center  of  the 
image,  we  consider  the  intensity  of  the  light  on  the  line  MN, 


114  OPTICALLY   UNIAXIAL   CRYSTALS 

we  observe  a  continuous  alternation  of  bright  and  dark.  And 
therewith,  as  the  distance  from  the  center  increases,  the  dis- 
tance between  the  light  minima  and  light  maxima  always  be- 
comes less;  because  with  greater  obliqueness  of  the  rays  to  the 
optic  axis  there  is  an  increase  not  only  in  the  difference  in 
velocity  of  the  two  interfering  rays  but  also  in  the  length  of 
path  they  must  travel  in  the  crystal,  so  that,  in  the  case  of 
the  rays  that  form  larger  angles  with  the  optic  axis,  the 
same  difference  in  the  inclination  corresponds  to  a  greater 
difference  in  the  retardation  than  in  the  case  of  those  standing 
at  smaller  angles  to  the  axis. 

Since  optically  uniaxial  crystals  behave  absolutely  the*same 
toward  all  light  rays  that  include  equal  angles  with  the  optic 
axis,  then  in  the  image  the  same  maxima  and  minima  of  light 
must  in  all  directions  be  present  at  the  same  distance  from  the 
center;  for  what  applies  to  the  principal  section  MN,  Fig,  59, 
must  apply  in  wholly  the  same  way  to  every  other  principal 
section,  standing  at  any- other  angle  to  PP'.  The  center  of  the 
field  must  accordingly  be  surrounded  by  bright  and  dark  rings 
of  exactly  circular  form.  The  bright  rings  can,  however,  not 
everywhere  have  the  same  intensity,  because  their  intensity 
corresponds  to  the  sum  of  two  variable  amplitudes;  these  ampli- 
tudes, as  follows  from  .Fig's  31-33  on  page  61,  are  at  their 
maximum  for  the  principal  sections  diagonal  to  A  A'  and  PP', 
diminishing  on  either  side;  and  for  the  two  principal  sections 
parallel  to  A  A '  and  PP'  they  become  zero.  In  consequence  of 
this  a  dark  cross  must  appear,  with  its  arms  parallel  to  A  A' 
and  PP';  — and  this  same  conclusion  also  follows  directly  from 
the  consideration  that,  for  any  inclination  within  one  of  these 
two  planes,  only  one  ray  is  formed  (because  the  component  per- 
pendicular to  it  is  zero),  and  this  ray  in  one  of  the  two  nicols 
totally  reflected.  All  phenomena  observed  on  a  circle  described 
about  the  center  correspond,  as  it  were,  to  those  that,  in  parallel 
polarized  light,  a  plate  of  the  same  thickness  and  inclination  to 
the  optic  axis  exhibits  in  succession  when  it  is  rotated  360°;  the 


BEHAVIOR   IN   THE   POLARIZATION   APPARATUS  115 

places  where  the  circle  intersects  the  arms  of  the  dark  cross 
correspond  exactly  to  the  four  positions  of  darkness  of  such  a 
plate.  Figure  59  accordingly  represents  the  interference-figure 
that  we  observe  with  crossed  nicols  when  the  instrument  is 
illuminated  with  monochromatic  light. 

If,  however,  for  the  illumination  we  choose  light  of  another 
color,  e.g.  a  color  having  greater  wave  length,  then,  obviously,  if 
the  birefringence  for  this  color  is  just  about  as  high  as  for  the 
last,  a  greater  inclination  than  before  will  be  necessary,  of  the 
rays  to  the  optic  axis,  in  order  to  give  the  rays  a  path  difference  of 
one  whole — now  greater  —  wave  length;  so  the  distance  of  the 
first  dark  ring  from  the  center,  and  likewise  that  of  the  succeed- 
ing rings  from  the  first, will  be  greater  than  with  the  previouscolor. 
The  less  the  wave  length  of  the  light  used  for  illuminating,  the 
narrower  will  be  the  rings  we  see  in  the  polariscope;  and  on 
the  other  hand,  the  greater  the  wave  length,  the  wider  the  rings. 

Hence  if,  instead  of  monochromatic,  we  use  white  light,  by 
causing  the  light  from  a  brightly  illuminated  part  of  the  sky 
to  be  reflected  from  the  mirror  of  the  conoscope  into  interior 
of  the  instrument,  those  rays  that  traverse  the  crystal  at  a 
certain  inclination  to  its  axis  will  so  interfere  that  for  one  defi- 
nite color  the  difference  of  path  amounts  exactly  to  X;  conse- 
quently, between  crossed  nicols  this  color  is  extinguished,  while 
the  others  are  weakened  the  less,  the  more  their  wave  length 
differs  from  that  of  the  color  in  question.  After  the  annihilation 
of  a  certain  color,  therefore,  the  light  appearing  at  the  point  in 
question,  of  the  image,  will  no  longer  exhibit  white,  but  a  com- 
posite color.  This  color  will  be  the  same  for  all  rays  having  the 
same  inclination  to  the  optic  axis  of  the  crystal;  accordingly,  all 
points  of  the  interference-figure  that  are  equally  distant  from 
the  center  will  exhibit  the  same  color,  but  all  those  of  different 
distance  different  color.  Thus,  with  crossed  nicols  there  appear 
colored  rings  intersected  by  a  black  cross.  (See  Plate  II,  Fig.  i.) 

When  the  nicols  are  parallel,  the  rays  interfere  with  the 
opposite  difference  of  path,  i.e.  the  same  with  which  they  emerge 


Il6  OPTICALLY   UNI  AXIAL   CRYSTALS 

from  the  crystal;  consequently,  instead  of  a  black,  a  white  cross 
appears,  and  at  every  distance  from  the  center  exactly  that 
color  is  extinguished  that  with  crossed  nicols  has  its  maximum 
intensity;  accordingly,  the  colored  rings  of  the  interference-figure 
with  the  black  cross  are  colored  exactly  in  complement  to  those 
of  the  same  diameter  in  the  figure  with  the  white  cross. 

The  curved  lines  of  an  interference-figure,  which  exhibit 
the  same  color  at  all  points,  are  called  isochromatic  curves; 
these  curves  for  an  optically  uniaxial  plate  perpendicular  to  the 
axis  are  thus  exact  circles,  whose  common  center  corresponds  to 
the  axis.  By  our  seeing  through  two  parallel  faces  of  a  crystal 
the  circular  isochromatic  curves  with  the  (with  _L  nicols)  dark 
cross  we  not  only  —  since  only  the  optically  uniaxial  exhibit 
this  phenomenon  —  recognize  the  crystal  as  optically  uniaxial, 
but,  in  addition,  the  direction  of  its  optic  axis  is  determined  as 
normal  to  the  pair  of  faces  looked  through. 

Since,  in  the  behavior  of  the  crystal  toward  rays  of  light,  all 
principal  sections  intersecting  one  another  in  the  optic  axis  are 
absolutely  equivalent,  the  interference-figure  yielded  in  the  cono- 
scope  by  such  a  plate  must  remain  absolutely  unaltered  when 
the  plate  is  rotated  in  its  own  plane.  Only  the  rotation  of  a 
nicol  can  alter  it;  and  this  rotation,  when  it  amounts  to  90°, 
transforms  the  previous  interference-figure  into  the  comple- 
mentary one  —  that  with  the  white  cross. 

The  colors  exhibited  by  the  interference-figure  represented  in 
Fig.  i,  Plate  II,  are  obviously  the  same*  as  were  considered 
in  detail  on  page  63  et  seq. ;  for  if  we  advance  from  the  center  of 
the  figure  to  the  edge,  we  arrive  at  the  points  of  convergence  of 
rays  that  experience  in  the  crystal  an  amount  of  double  refrac- 
tion beginning  with  zero  (at  the  center)  and  continuously  in- 
creasing; in  other  words,  we  observe  side  by  side  the  same 
phenomena  that  a  series  of  plates  of  the  crystal  would  exhibit  to 
us  in  succession,  should  we  begin  with  a  plate  cut  perpendicular 

*  With  those  differences,  usually  small,  that  result  from  the  inequality  of  the 
birefringence  for  different  colors.     (Cf.  p.  69  and  pp.  108-109.) 


BEHAVIOR   IN   THE   POLARIZATION   APPARATUS 


117 


to  the  axis  and  then  follow  with  such  as  had  an  increasing  incli- 
nation of  their  normal,  to  the  axis,  combined  with  an  increase 
in  thickness  corresponding  to  this  inclination.  If  within  the 
field  of  the  conoscope  there  fall  rays  that  in  the  crystal  were 
retarded  a  larger  number  of  wave  lengths  as  compared  with 
each  other,  the  white  of  a  higher  order  naturally  appears.  Thus, 
while  in  monochromatic  light  a  crystal  plate  of  high  birefrin- 
gence exhibits  the  interference-figure  represented  in  Fig.  60,  i.e. 
a  series  of  bright  and  dark  rings 
which  become  narrower  and  nar- 
rower from  the  center  out  to  the 
edge  of  the  field  or  until  they  are 
so  fine  that  they  can  no  longer  be 
distinguished,  the  same  plate  in 
white  light  yields  only  a  limited 
number  of  rings,  since  those  hav- 
ing the  colors  of  even  the  fifth 
and  sixth  orders  differ  very  little 
from  white,  and  since  beyond 
these  rings  a  gradual  transition 
takes  place  into  the  uniform 
light  gray  of  the  field.  Here,  therefore,  a  circle  described  about 
the  center  corresponds,  as  it  were,  to  the  rotation  in  its  own 
plane  of  a  plate  having  such  high  birefringence  that  between 
the  four  positions  of  darkness  it  exhibits  only  a  simple  light- 
ening, without  color. 

Up  to  the  present  one  circumstance  has  been  disregarded ; 
namely,  the  thickness  of  the  crystal  plate.  If  instead  of  the  plate 
hitherto  considered,  which  we  always  supposed  of  equal  thick- 
ness, we  take  another  plate  of  the  same  optically  uniaxial  sub- 
stance, likewise  perpendicular  to  the  axis  but  only  half  as  thick, 
the  two  rays  arising  from  one  in  this  plate  by  the  double 
refraction  will,  when  they  have  the  same  inclination  to  the  axis 
as  before,  travel  only  half  as  far  in  the  crystal,  and  accordingly 
the  relative  retardation  will  be  only  half  as  great.  The  same 


Fig.  60. 


Il8  OPTICALLY   UNIAXIAL   CRYSTALS 

path  difference,  therefore,  which  requires  the  same  interference, 
can  arise  only  when  we  come  to  rays  having  a  far  greater  incli- 
nation to  the  axis;  thus,  for  example,  in  monochromatic  light  the 
first  dark  ring  of  the  interference-figure  can  be  formed  only 
at  a  much  greater  distance  from  the  center;  and  this  applies  in 
the  case  of  white  light  to  each  of  the  rings  of  color.  The 
isochromatic  curves  exhibited  by  a  uniaxial  crystal  will  accord- 
ingly stand  the  farther  apart,  the  thinner  the  plate  investigated; 
the  closer  together,  the  thicker  it  is  chosen. 

Finally,  if  we  compare  the  color  rings  exhibited  by  equally 
thick  plates  of  different  substances,  we  find  that,  in  consequence 
of  the  inequality  of  the  birefringence  of  different  bodies,  the 
rings  differ  in  their  width.  The  explanation  is  that  in  a  crystal 
of  lower  birefringence  —  in  which  the  difference  between  the 
velocity  of  the  ordinary  and  of  the  extraordinary  ray  is  slight  — 
the  rays,  in  order  to  each  acquire  a  path  difference  of  one  wave 
length,  will  have  to  be  inclined  to  the  optic  axis  at  a  greater 
angle  than  in  a  crystal  of  higher  birefringence;  so  that  with 
equal  plate  thickness  the  color  rings  of  the  interference-figure 
must  be  wider*  As  an  example  of  a  substance  having  very  high 
birefringence,  whose  plates  accordingly,  unless  very  thin,  always 
exhibit  narrow  rings,  calcite  may  be  mentioned.  When  the 
double  refraction  of  an  optically  uniaxial  substance  is  very 
weak,  and  at  the  same  time  the  inequality  of  its  strength  for 
the  different  colors  great  enough  to  be  noticeable  (as  is  the  case 
e.g.  with  the  mineral  apophyllite) ,  there  results  considerable 
deviation  of  the  colors  in  the  rings  of  the  interference-figure 
from  the  normal  scale  of  the  colors  of  the  first,  second,  third, 
etc.,  orders.  For,  if  the  first  dark  ring  for  red  appears  at  a  cer- 
tain distance  from  the  center,  and  if  that  for  blue  —  which  by 
reason  of  the  lesser  wave  length  would,  with  equal  birefringence, 
be  formed  nearer  the  center  —  appears,  in  consequence  of  the 

*  Hence  it  is  seen  that  the  width  of  the  rings  may  serve  to  determine  the 
refractive  index  of  the  extraordinary  ray,  if  that  of  the  ordinary  is  known.  (Con- 
cerning this  method  see  Zeitschr.f.  Kryst.  7,  394.) 


BEHAVIOR   IN   THE   POLARIZATION    APPARATUS  119 

birefringence  for  this  color  being  lower  and  the  rings  therefore 
wider,  at  the  same  place  as  that  for  red,  then  here  red  and  blue 
must  be  annihilated,  and  yellow  therefore  arise;  while  some- 
what farther  toward  the  center,  where  otherwise  red  is  extin- 
guished and  blue  appears,  the  red  will  now  have  an  intensity 
such  that  the  mixing  here  of  the  red  and  the  blue  gives  violet. 
Thus  color  rings  arise  in  which  the  red  is  in  most  cases  wholly 
lacking;  and  in  certain  crystals  of  the  mineral  named  the  dark 
rings  for  all  the  colors  so  nearly  coincide  that  to  all  appearance 
only  white  and  black  rings  arise  (wherefore  this  variety  has  been 
named  leucocyclite) .  Some  crystals  of  apophyllite  belong  with 
the  substances  already  mentioned,  on  page  109,  as  having  such 
slight  birefringence  that  for  the  colors  forming  one  end  of  the 
spectrum  they  are  positively  doubly  refracting,  for  those  of  the 
other  end  negatively  doubly  refracting:  with  the  former  colors 
the  ordinary  ray,  with  the  latter  the  extraordinary,  is  trans- 
mitted the  faster,  so  that  there  is  an  intermediate  color  for  which 
the  crystals  are  singly  refracting  (without  for  that  reason  belong- 
ing with  the  optically  isotropic  crystals,  since  these  are  singly 
refracting  for  all  colors,  but  the  crystals  in  question,  for  all  the 
remaining  colors,  really  uniaxial). 

Plates  parallel  or  oblique  to  the  optic  axis,  when  they  are 
investigated  in  convergent  light,  can,  if  this  is  white,  according 
to  the  exposition  on  page  70  yield  color  phenomena  only  in  case 
their  thickness  is  very  slight.  Any  such  plate  will  produce  at  the 
center  of  the  field  the  same  color  that  one  observes  through  it 
in  parallel  light.  The  rays  that  on  the  other  hand  pass  through 
the  plate  in  an  inclined  direction,  although  they  will  indeed  all 
travel  an  increasing  length  of  path  in  the  crystal,  will  not  acquire 
a  difference  of  path  increasing  in  the  same  way;  since  in  certain 
directions,  namely,  where  they  approach  the  optic  axis,  the 
difference  in  the  velocity  of  the  two  rays  continuously  dimin- 
ishes, wherefore  in  spite  of  the  increasing  thickness  traversed 
their  path  difference  also  becomes  less.  In  directions  perpen- 
dicular to  these  inclined  directions,  when  the  plate  is  cut  parallel 


120  OPTICALLY    UNIAXIAL    CRYSTALS 

to  the  optic  axis,  the  difference  in  velocity  will  not  vary;  so  with 
growing  inclination,  and  consequent  increase  of  the  thickness  trav- 
ersed, the  difference  of  path  also  will  increase.  In  intermediate 
directions  the  two  effects  will  neutralize  each  other,  and  the  path 
difference  will  remain  the  same  for  all  inclinations  of  the  rays  to 
the  normal  of  the  plate.  In  homogeneous  light,  systems  of  dark 
and  bright  stripes  thus  arise,  which  have  in  general  hyperbolic 
form;  in  white  light  the  stripes  are  colored,  but  are  visible  only 
when  the  plate  is  very  thin,  since  with  greater  thickness  the 
white  of  a  higher  order  results.  Only  that  phenomenon  is  of 
special  practical  interest  in  crystallography,  that  is  exhibited  by 
a  plate  inclined  at  no  great  angle  (not  over  35°  or  40°)  to  the 
plane  standing  normal  to  the  optic  axis;  for  with  such  a  plate, 
in  convergent  light  and  in  a  polariscope  having  a  large  field, 
rays  still  converge  within  the  field  that  pass  through  the  crystal 
along  the  axis.  Near  the  edge  of  the  field,  in  some  direction, 
the  observer  will  accordingly  behold  the  interference-figure  of 
the  axis  —  the  black  cro^s  with  the  color  rings  (although  the 
latter  are  no  longer  exactly  circular).  In  this  direction  there- 
fore, inclined  at  an  acute  angle  to  the  normal  of  the  plate,  there 
lies  the  optic  axis,  for  finding  which  the  phenomenon  in  question 
may  serve. 

The  conversion  of  the  microscope  into  a  conoscope  (see  p. 
80)  is  of  special  advantage  for  determining  an  optically  uniaxial 
crystal  as  such  when,  for  example,  in  a  rock  section  (see  foot- 
note, p.  108)  there  are  visible  a  number  of  differently  oriented 
sections  of  a  mineral  most  of  which  appear  doubly  refracting, 
while  on  the  rotation  between  crossed  nicols  some  remain  dark. 
If  the  sections  in  question  belong  to  an  optically  uniaxial  min- 
eral, those  last-mentioned  must  be  such  whose  optic  axis  happens 
to  be  directed  perpendicular  to  the  plane  of  the  rock  section, 
and  must  therefore,  when  convergent  light  is  employed,  exhibit 
the  above-described  axial  figure,  the  colored  rings  with  the 
black  cross. 


OPTICAL   INDEX-SURFACE  121 


OPTICALLY  BIAXIAL  CRYSTALS 

DEDUCTION  OF  THE  OPTICAL  PROPERTIES  OF  CRYSTALS 
FROM  A  SURFACE  OF  REFERENCE  (OPTICAL  INDEX- 
SURFACE  OR  INDICATRIX) 

On  page  95  et  seq.  the  optical  properties  of  the  uniaxial  crystals 
were  deduced  from  their  ray-surface;  i.e.  from  the  double  sur- 
face whose  radii  vectores  correspond  to  the  velocities  of  the  rays 
transmitted  parallel  to  these  radii.  Except  in  the  optic-axial 
direction  each  radius  vector  intersects  ~the  surface  twice;  that 
is  to  say,  along  the  same  path,  but  with  different  velocity,  two 
rays  are  transmitted  whose  ray-fronts  are  respectively  the  tan- 
gential planes  of  the  double  surface  at  the  two  points  of  inter- 
section. As  already  mentioned  on  page  96,  and  as  may  be  seen 
from  Fig.  51  et  seq.,  the  front  of  the  ordinary  ray  always  stands 
perpendicular  to  the  ray;  that  of  the  extraordinary,  however,  in 
general  not  perpendicular,  since  the  tangent  of  an  ellipse  is  normal 
to  the  radius  vector  only  when  this  coincides  with  one  of  the 
principal  axes  of  the  ellipse.  While,  therefore,  for  the  ordinary 
ray  the  front-normal  (called  also  the  " wave-normal")  coincides 
with  the  ray,  with  the  extraordinary  such  is  in  general  not  the  case. « 

In  Fig.  6 1  (p.  122),  then,  let  there  be  represented  the  principal 
section  of  the  ray-surface  of  a  uniaxial  crystal.  If  Or  be  any 
radius  vector  of  the  wave-surface  of  the  extraordinary  ray,  rV 
the  tangent  to  the  ellipse  at  the  point  r,  and  finally  OR  the 
radius  vector  parallel  to  this  tangent,  then  the  tangent  to  the 
ellipse  at  R,  namely  RV,  is  parallel  to  Or;  Or  and  OR  are  then 
two  so-called  "  conjugate  radii  "  of  the  ellipse,  for  which  the 
rule  holds  good,  that  the  area  of  the  parallelogram  ORVr  is 
constant  and  equal  to  that  of  the  parallelogram  OX  •  OZ,  — i.e. 
equal  to  the  product  of  the  two  semi-axes  of  the  ellipse.  If  we 
draw  RN  perpendicular  to  Or,  then,  as  is  easily  seen,  the  area 
of  ORVr  is  equal  also  to  the  rectangle  we  obtain  if  from  O  and 
r  we  let  fall  perpendiculars  to  the  tangent  RV,  — i.e.  equal  to 


122 


OPTICALLY  BIAXIAL  CRYSTALS 


the  right-angled  parallelogram  Or  •  RN.  Thus,  for  every  direc- 
tion of  the  ray,  the  area,  Or  •  RN  is  equal  to  the  constant  product 
OX  •  OZ;  consequently 

__OX.QZ  _      i 

RN      ~  RN 

So  the  transmission  velocity  of  any  extraordinary  ray,  i.e.  the 
length  Or,  is  proportional  to  the  reciprocal  length  of  the  straight 
line  that  stands  perpendicular  both  to  the  ellipse  (consequently 
v  also  to  the  spheroid)  at  the  point 

R,  and  to  the  ray  itself.  But 
this  line  RN,  according  to  our 
previous  assumption  (see  foot- 
note, p.  89),  is  at  the  same  time 
the  vibration  direction  of  the 
ray  Or;  for  the  polarization 
plane  of  this  ray  is  the  plane 
passed  normal  to  RN  through 
the  transmission  line  Or  of  the 
ray  and  perpendicular  to  the 
principal  section. 
THE  POINT  R  ON  THE  ROTATION  ELLIPSOID  THUS  DETERMINES 

ABSOLUTELY  THE  TRANSMISSION  DIRECTION,  THE  VELOCITY,  AND 
THE  POLARIZATION  OF  AN  EXTRAORDINARY  RAY  Or.  One  has  Only 

to  draw  the  normal  to  the  surface  at  the  point  in  question,  i.e. 
the  normal  to  the  tangential  plane  at  this  point,  and  then  obtains 
the  direction  of  the  ray  proper  to  the  point  by  dropping  a  per- 
pendicular from  the  center  to  this  normal.  These  two  lines 
define  all  the  three  above-named  elements,  the  ray  characters, 
marking  the  properties  of  the  ray. 

But,  for  any  radius  vector  Or  of  a  rotation  ellipsoid  there  are 
two  normals  to  the  surface  that  stand  at  the  same  time  perpen- 
dicular to  the  radius  vector.  One  of  them,  of  course,  is  RN^ 
The  other  is  the  perpendicular  erected  at  O  to  the  principal  sec- 
tion represented  in  Fig.  61;  this  perpendicular  always  lies  in 
the  equatorial  plane  of  the  ellipsoid  and  therefore  has  the  con- 


Fig.  61. 


OPTICAL   INDEX-SURFACE  123 

stant  length  OX,  however  Or  be  inclined  in  the  principal  section. 
Now  this  second  normal  obviously  bears  the  same  relation  to 
the  ordinary  ray  Os  as  does  RN  to  the  extraordinary  Or;  for 
according  to  our  assumption  it  is  the  vibration  direction  of  the 
former  ray;  and  since  its  length  isDX.  but  the  product  OX  •  OZ 

constant  (or  =  i)  and  OZ  therefore  proportional  to  -— ,  the  vel- 

OX 

ocity  of  the  ray  proper  to  this  second  normal  is  inversely  propor- 
tional to  its  length  OX. 

THE  CHARACTERS  OF  THE  ORDINARY,  AS  OF  THE  EXTRAORDI- 
NARY, RAY  OF  ANY  DIRECTION  ARE  DEDUCIBLE  THEREFORE  FROM 
A  SINGLE  SURFACE,  A  SPHEROID  THE  LENGTH  OF  WHOSE  ROTATION 
AXIS  STANDS  TO  THE  DIAMETER  OF  ITS  EQUATOR,  OR  CIRCULAR 
SECTION,*  IN  THE  RATIO  OF  THE  RECIPROCAL  VALUES  OF  THE 
TRANSMISSION  VELOCITIES  OF  LIGHT- VIBRATIONS  THAT  TAKE  PLACE 
PARALLEL  AND  PERPENDICULAR  TO  THE  AXIS;  'I.e.  (according  to 
p.  IQO)  IN  THE  RATIO  OF  THE  TWO  PRINCIPAL  REFRACTIVE  INDICES 
£  AND  (JL>.  THIS  SURFACE  SHALL  THEREFORE  BE  CALLED  THE  OPTI- 
CAL INDEX-SURFACE  OR  "INDICATRIX".  IT  HAS  FOR  A  NEGATIVE 
UNIAXIAL  CRYSTAL  THE  FORM  OF  A  SPHEROID  WHOSE  ROTATION 
AXIS  CORRESPONDS  TO  THE  SHORTEST  DIAMETER;  FOR  A  CRYSTAL 
WITH  POSITIVE  DOUBLE  REFRACTION,  ON  THE  OTHER  HAND,  I.e. 
WHEN  e  >  W,  IT  HAS  THE  FORM  OF  THE  SURFACE  THAT  IS  GEN- 
ERATED BY  THE  ROTATION  OF  AN  ELLIPSE  ABOUT  ITS  MAJOR  AXIS. 

From  this  surface,  which  accordingly  is  defined  by  the  two 
principal  refractive  indices  of  the  crystal  and  therefore  has  for  a 
different  color  a  different  axial  ratio,  the  optical  properties  of  the 
crystal  in  any  direction  are  deduced  as  follows:  — 

To  the  diameter  parallel  to  the  direction  in  question  there  in 
general  correspond  two  points  on  the  surface,  the  normal  to 
the  surface  at  these  points  standing  at  the  same  time  perpendic- 
ular to  that  diameter;  along  the  direction  of  the  latter,  there- 
fore, two  rays  are  transmitted  whose  vibration  directions  are 

*  Thus  will  be  designated  the  section  through  the  center  perpendicular  to  the 
rotation  axis:  this  section  (like  every  one  parallel  to  it)  obviously  has  the  form  of 
a  circle. 


124 


OPTICALLY   BIAXIAL   CRYSTALS 


those  two  normals  and  whose  velocities  are  to  each  other  in- 
versely as  the  lengths  the  ray  intercepts  on  the  two  normals. 
Only  one  diameter,  the  rotation  (i.e.  optic)  axis,  is  different  in 
this  respect;  to  it  there  correspond  an  infinity  of  points  on  the 
index-surface;  for  the  normal  to  the  surface  at  every  point  of  the 
equatorial  plane  stands  perpendicular  to  this  diameter.  Parallel 
to  the  axis,  therefore,  rays  are  transmitted  whose  vibrations  take 
place  in  all  possible  directions  perpendicular  to  the  axis.  In  a 

word:  to  every  diameter  (ray) 
there  correspond  two  vibration 
directions;  to  the  axis,  only,  an 
infinity  of  vibration  directions. 
Conversely,  to  every  point  of 
the  index-surface  of  a  uniaxial 
crystal  there  in  general  corre- 
4  spends  one  ray.  The  direction 
of  this  ray  is  that  of  the  diam- 
eter intersecting  at  right  angles 
the  normal  erected  on  the  sur- 
face at  the  point  in  question;  the 
transmission  velocity  is  inversely  proportional  to  the  intercept 
on  this  normal  between  the  surface  and  the  ray;  the  polariza- 
tion plane  stands  perpendicular  to  this  same  normal,  the  normal 
being  the  vibration  direction  of  the  ray.  If  the  given  point  lies 
neither  in  the  equatorial  plane  of  the  index-surface  nor  at  one 
end  of  the  optic  axis,  the  corresponding  vibration  direction  lies 
in  the  plane  passing  through  axis  and  ray,  i.e.  in  the  principal 
section;  so  the  ray  is  extraordinary.  If  the  point  lies  at  one  end 
of  the  axis,  e.g.  if  in  Fig.  61  R  coincides  with  Z,  then  the  direc- 
tion of  the  ray  proper  to  it  is  indefinite,  because  the  normal  to 
the  surface  then  passes  through  the  center,  wherefore  the  points 
N  and  O  likewise  coincide;  to  the  point  Z,  therefore,  there  cor- 
respond all  the  extraordinary  rays  transmitted  in  the  equatorial 
plane,  which  all  have  the  vibration  direction  OZ  and  the  veloc- 
ity OX.  If  on  the  other  hand  the  given  point  lies  on  the  "  cir- 


Fig.  61. 


OPTICAL   INDEX-SURFACE  125 

cular  section  "  of  the  index-surface,  the  normal  erected  here  on 
the  surface  likewise  passes  through  the  center;  so  that,  in  this 
case  also,  the  direction  of  the  ray  is  indefinite.  That  is  to  say, 
to  such  a  point  there  correspond  an  infinity  of  rays  which  are 
transmitted  in  the  plane  that  stands  perpendicular  to  the  normal 
erected  at  the  point;  the  rays  all  vibrate  normal  to  this  plane, 
and  therefore  they  have  the  constant  velocity  OZ;  for  this 
quantity  is  inversely  proportional  to  the  length  OX  oi  the  nor- 
mals. These  rays,  since  their  vibration  direction  is  perpendic- 
ular to  the  principal  section,  are  all  ordinary.  While,  therefore, 
to  the  two  points  at  the  ends  of  the  axis  there  correspond  an 
infinity  of  extraordinary  rays  transmitted  in  the  equatorial  sec- 
tion, there  belong  to  every  point  of  the  equatorial  section  an 
infinity  of  ordinary  rays  transmitted  in  one  principal  section; 
and  since  there  are  an  infinity  of  points  of  the  latter  kind,  the 
index-surface  has  an  infinity  of  equivalent  principal  sections  cor- 
responding to  them,  which  all  intersect  one  another  in  the  axis. 

THE  ANALOGOUS  OPTICAL  SURFACE  OF  REFERENCE  FOR  SINGLY 
REFRACTING  BODIES  IS  A  SPHERE.  For  with  SUch  bodies  OJ  =  £' 

in  other  words,  the  refractive  index  is  the  same  in  all  directions. 
The  normal  erected  on  the  index-surface  at  its  every  point  then 
passes  through  the  center,  and  the  direction  of  the  ray  is  indefi- 
nite; that  is  to  say,  all  rays  of  the  vibration  direction  in  question, 
however  they  are  transmitted  in  the  plane  perpendicular  to  that 
direction,  have  the  same  velocity  and  hence  are  ordinary.  But 
since  the  normals  at  all  points  of  the  surface  have  equal  length, 
the  ray  velocity  is  equal  in  all  planes  of  the  crystal,  wherefore 
in  such  a  crystal  there  can  be  only  ordinary  rays. 

Just  as  the  index-surface  of  the  singly  refracting  crystals 
presents  that  special  case  of  the  surface  for  uniaxial  crystals  *  in 

*  As  was  mentioned  on  p.  109  etseq.,  there  are  some  uniaxial  crystals  that  for 
a  certain  color  are  singly  refracting.  Here,  then,  we  have  this  special  case;  in 
other  words,  the  index-surface  for  light  of  the  color  in  question  is  a  sphere,  but  for 
the  remaining  colors  a  rotation  ellipsoid,  in  part  prolate  (with  negative  double 
refraction),  in  part  oblate  (with  negative  double  refraction).  But  with  the  major- 
ity of  optically  uniaxial  crystals  the  index-surfaces  for  the  different  colors  are  of 
one  and  the  same  kind,  differing  only  in  their  axial  ratio. 


126  OPTICALLY   BIAXIAL  CRYSTALS 

which  the  diameter  parallel  to  the  axis  is  e'qual  to  the  diameter 
perpendicular  to  the  axis,  so  again  might  the  uniaxial  index-sur- 
face be  conceived  of  as  a  special  case  of  one  that  is  still  more  gen- 
eral; namely,  of  an  index-surface  whose  diameter  is  different  in  all 
three  dimensions  of  space.  This  surface  were  then  a  so-called 
triaxial  ellipsoid,  with  three  mutually  perpendicular,  unequal 
principal  axes;  and  the  lengths  of  the  three  principal  axes  must 
be  proportional  to  the  refractive  indices  of  the  rays  that  corres- 
pond to  their  extremities.  In  such  an  ellipsoid,  as  is  pictured 
in  Fig.  62  on  the  opposite  page,  the  three  principal  axes  stand 
perpendicular  to  the  surface  at  their  extremities;  so  the  normals 
to  the  surface  at  these  six  points  (and  at  no  others)  pass  through 
the  center  of  the  ellipsoid.  To  such  a  point,  therefore,  exactly 
as  to  a  point  on  the  equatorial  circle  of  the  uniaxial  index- 
surface,  there  correspond  an  infinity  of  ordinary  rays  lying  in 
the  principal  section  that  stands  perpendicular  to  the  normal 
erected  at  the  point,  and  having  the  same  velocity  and  vibration 
direction.  But,  while  with  the  uniaxial  index-surface  there 
exist  an  infinity  of  such  principal  sections,  here  only  three  are 
possible,  the  three  that  stand  perpendicular  to  the  three  prin- 
cipal axes  of  the  ellipsoid;  and,  since  the  transmission  velocity 
of  the  ordinary  rays  within  such  a  principal  section  is  inversely 
proportional  to  the  semi-axis  standing  perpendicular  to  it,  but 
the  length  of  that  semi-axis  always  different,  these  rays  have  in 
each  of  the  three  principal  sections  a  different  refractive  index. 
These  three  refractive  indices,  which  accordingly  determine  the 
form  of  the  index-surface,  are  called  the  three  principal  refrac- 
tive indices;  the  respective  vibration  directions  of  these  three 
kinds  of  ordinary  rays,  which  therefore  are  nothing  else  than  the 
directions  of  the  three  principal  axes  of  the  ellipsoid,  are  known 
as  the  principal  vibration  directions. 

The  threb  principal  refractive  indices  will  be  denoted  by 
a,  /?,  /-;  an3  of  these  a  shall  be  the  smallest  index,  /?  the  so- 
called  intermediate  (it  may  lie  the  closer  either  to  a  or  to  f), 
f  the  largest.  If  with  its  semi-axes  of  the  lengths  OX  =  a, 


OPTICAL   INDEX-SURFACE 


I27 


OF=  /?,  OZ  =  f  we  construct  the  triaxial  ellipsoid  represented 
in  Fig.  62,  then  by  each  of  the  three  principal  optic  sections 
XZXZ,  XYXY,  YZYZ  this  surface  will  be  intersected  in  a 
different  ellipse;  and  each  of  those  three  sections  divides  the 
whole  form  into  two  equal  and  opposite  halves.  They  are 
"  planes  of  symmetry  "  (see  p.  28)  of  the  surface,  something 
which  applies  to  no  further  planes.  A  random  section  through 
the  center  has  in  general  likewise  the  form 
of  an  ellipse,  but  of  one  whose  axial  ratio 
varies  with  orientation  of  the  intersecting 
plane ;  two  planes  only,  besides  the  three 
principal  sections,  are  of  special  impor- 
tance, they  being  so  for  the  reason  that 
they  intersect  the  surface  not  in  an  ellipse 
but  in  a  circle.  That  such  is  the  case  is 
brought  out  by  a  consideration  of  the 
following:  The  principal  section  XYXY 
of  the  triaxial  ellipsoid  Fig.  62  is  an  ellipse 
whose  major  and  minor  axes  are  to  each 
other  as  the,  Intermediate  to  the  smallest 
refractive  index;  a  section  taken  through 
YY  but  inclined  to  the  plane  XYXY 
is  obviously  (because  of  the  symmetry  of 
the  form)  likewise  an  ellipse,  whose  one 
axis  is  YY  and  whose  other  axis  is  a  radius_ 
vector  of  the  principal-section  ellipse  XZXZ-,  but  these  radii  vec- 
tores  have,  according  to  the  inclination,  all  possible  values  between 
OX  (=  the  smallest  refractive  index  a)  and  OZ  (=  the  largest 
7-);  consequently,  somewhere  between,  there  must  be  a  radius 
vector,  OCt,  that  is  equal  to  the  intermediate  refractive  index 
/?;  the  intersection  curve  corresponding  to  this  radius  vector, 
namely  YC1YCV  has  equal  axes,  or  in  other  words  is  a  circle. 
This  form  is  approached  by  the  ellipses  for  lesser  inclinations 
as  their  inclination  increases,  the  minor  axis  always  becoming 
more  like  the  major,  the  more  it  approaches  the  direction  OCl; 


128 


OPTICALLY    BIAXIAL   CRYSTALS 


while  for  greater  inclinations  the  major  axis  is  the  one  lying  in 
the  plane  XZXZ,  and  YY  now  becomes  the  minor.  Finally, 
after  a  rotation  of  90°,  i.e.  for  the  sectibn  FZFZ,  the  variable 
axis  has  its  greatest  length.  This  axis  then  grows  smaller 
again,  on  the  other  side  of  FZFZ;  and  in  the  direction  OC2, 
which  — because  of  the  symmetry  of  the  triaxial  ellipsoid  with 
reference  to  the  three  principal  sections  —  forms  with  OZ  the 

same  angle  as  does  OCV  it  attains  again 
the  value  OF.  Accordingly,  the  triaxial 
ellipsoid  has  two  circular  sections, 


YC1YC1  and 


FC2FC2,  which  intersect 


each  other  in  the  intermediate  axis  at  an 
angle  that  is  bisected  by  the  two  prin- 
cipal sections  passing  through  this  axis. 
The  sections  through  XX  are  all  ellipses 
whose  minor  axis  is  a  and  whose  major 
varies  from  ft  to  ?-,  while  the  sections 
through  ZZ  are  ellipses  having  the 
major  axis  j-  and  a  minor  axis  varying 
from  a  to  /?;  in  neither  case,  therefore, 
can  the  ellipses  ever  assume  the  form 
of  a  circle. 

Now  a  triaxial  ellipsoid,  whose  prop- 
erties are  described  in  the  foregoing, 
would  present  the  most  general  case  of  an 
optical  index-surface.  For  the  ellipsoid  would  pass  over  into  the 
index-surface  of  a  uniaxial  crystal,  i.e.  into  a  rotation  ellipsoid,  if 
two  of  its  principal  axes  became  of  equal  length;  into  that  of  a 
singly  refracting  crystal,  i.e.  a  sphere,  if  all  three  of  its  principal 
axes  assumed  the  same  value.  In  fact,  it  turns  out  that  if,  in- 
stead of  with  the  two  special  cases  last  mentioned,  one  starts  out 
with  such  an  ellipsoid  having  three  unequal  axes,  as  the  surface 
of  reference,  and  from  this  ellipsoid  derives  the  ray-surface 
in  the  same  way  as  was  done  with  the  uniaxial  crystals,  then 
for  this  ray-surface  there  results  a  form  corresponding  to  the 


RAY-SURFACE  129 

optical  properties  of  all  the  crystals  that  are  neither  singly 
refracting  nor  optically  uniaxial.  Since  the  index-surface  of 
such  crystals  has  two  circular  sections,  the  directions  normal  to 
these  sections  are  in  a  certain  sense  analogous  to  the  optic  axis 
of  the  uniaxial  crystals  (this  being  the  normal  to  the  single  cir- 
cular section  of  the  latter  crystals);  therefore  these  directions 
also  are  spoken  of  as  optic  axes,  *  and  the  crystals  whose  ray- 
surface  is  derived  from  a  triaxial  ellipsoid  as  their  index-surface 
are  known  as  optically  biaxial  crystals. 

RAY-SURFACE  OF  THE  OPTICALLY  BIAXIAL  CRYSTALS 

The  ray-surface  proper  to  an  index-surface  f  having  the  form 
of  a  triaxial  ellipsoid  is  obtained  if  for  every  point  of  this  index- 
surface  we  find  the  corresponding  ray;  i.e.  the  diameter  that 
stands  perpendicular  to  the  normal  erected  on  the  surface  at 
the  point  in  question:  to  this  ray  there  in  general  then  corre- 
sponds yet  a  second  point  on  the  index-surface,  the  normal  to 
that  surface  at  this  second  point  likewise  standing  perpendicular 
to  the  ray;  and  on  these  two  normals  the  ray  intercepts  two 
lengths  whose  reciprocal  values  are  the  velocities  of  the  two 
light-motions  that  vibrate  parallel  to  the  two  normals  respectively, 
these  motions  being  transmitted  along  the  direction  of  the 
diameter  in  question.  Parallel  to  every  radius  vector  of  the 
index-surface  we  thus  obtain  two  rays,  and  ultimately,  for  all 
directions,  a  double  surface;  of  this  surface,  the  form  of  its 
intersection  with  the  three  principal  sections  of  the  index- surface 
shall  first  be  considered. 

*  [Sometimes  designated  more  specifically  as  primary  optic  axes.  (Cf .  pp. 
132  and  142.)] 

f  This  surface  was  called  by  Cauchy  the  "ellipsoid  of  polarization";  by 
Billet,  Verdet,  and  others  the  "inverse  ellipsoid  (of  elasticity,  of  velocity)";  by 
MacCullagh  the  "ellipsoid  of  indices";  by  Stefan  the  "ellipsoid  of  equal 
work";  by  Kirchoff  the  "ellipsoid  of  elasticity";  finally,  by  Fletcher,  whom 
we  follow  here  in  essentials,  the  "indicatrix".  The  name  "index-surface"  has, 
besides,  been  applied  by  some  authors  to  another  surface  of  reference,  of  which  we 
have  here  no  need. 


I30 


OPTICALLY    BIAXIAL   CRYSTALS 


Let  us  begin  with  the  principal  section  XZXZ,  Fig.  62.  In 
this  section  there  are  transmitted  an  infinity  of  ordinary  rays, 
which  correspond  to  the  point  Y  of  the  index-surface;  for  the 
normal  to  the  latter  at  F,  it  being  the  normal  at  the  extremity 
of  one  of  the  principal  axes,  passes  through  the  center,  where- 
fore the  direction  of  the  rays  proper  to  it  is  indefinite.  These 
rays  have  the  vibration  direction  OY  and  therefore  are  trans- 
mitted with  the  velocity  proportional  to 
i//?.  So  if  about  O  we  describe  a  circle 
with  the  radius  OB,  proportional  to  this 
quantity,  we  obtain  the  locus  at  which  all 
ordinary  rays  from  O  have  arrived  in  the 
same  time;  and  this  circle  constitutes 
one  of  the  curves  in  which  the  ray-sur- 
face is  intersected  by  the  principal  section 
XZXZ.  (Cf.  Fig.  64.)  To  obtain  the 
other  intersection,  which  corresponds  to 
the  extraordinary  rays  transmitted  in  the 
same  plane,  we  begin  with  the  point  X  of 
the  index  surface.  To  this  point,  since 
it  likewise  is  an  extremity  of  a  prin- 
cipal axis,  there  correspond  an  infinity 
of  rays  in  the  planejDerpendicular  to  that 
axis,  the  plane  FZFZ;  these  rays  all  have 
the  vibration  direction  OX  and  therefore 
the  velocity  i /a,  which  quantity,  it  being  the  reciprocal  of  the 
smallest  index,  is  the  greatest  light  velocity  in  the  crystal  in 
question.  In  the  plane  YZYZ  the  ray-surface  accordingly  has 
a  section  of  circular  form  with  the  radius  OA  =  i/a.  (Cf.  Fig. 
65.)  Among  these  rays  belongs  the  ray  that  is  parallel  to  OZ; 
so  in  this  direction  we  obtain  a  second  point  of  the  ray-surface 
if  in  Fig.  64  we  lay  off  a  length  OA,  proportional  to  the  greatest 
light  velocity.  The  two  lengths  OA  and  OB  have  the  ratio, 
therefore,  of  the  intermediate  to  the  smallest  principal  refractive 
index.  Hence,  if  on  the  ellipse  XZ  of  the  index-surface  we  pass 


RAY-SURFACE 


out  from  X  over  to  adjacent  points,  we  obtain  e.g.  for  the  point 
R,  Fig.  63,  the  corresponding  ray,  Or,  in  the  usual  way;  this 
ray  has  the  vibration  direction  RN,  and  its  velocity  is  inversely 
proportional  to  the  length  RN.  But  RN,  the  distance  of  the 
tangent  to  the  ellipse  at  R  from  the  radius  vector  Or  parallel  to 
it,  is  obviously  greater  than  a  =  OX',  for  this  latter  length  too  is 
the  distance  of  a  tangent  (at  X)  from  the  radius  vector  parallel  to  it 


(OZ)  ,  and  altogether  the  least  that  exists.  Consequently  the  recip- 
rocal value  —  -r  is  less  than  the  length  OA  in  Fig.  64,  since  this 


was  made  proportional  to  i/a.  So  in  the  direction  parallel  to  Or 
we  have  to  lay  off  a  length  OH  (see  Fig.  64)  ,  correspondingly 
shorter  than  OA.  But  in  the  same  direction,  as  we  saw  on 
page  130,  there  is  transmitted  a  second  ray,  with  the  velocity  pro- 
portional to  i//?,  and  this  ray  correspgnds  to  the  point  Y  on  the 
index-surface;  accordingly  the  point  H  must  be  nearer  to  the 
point  B  on  OH  than  is  A  to  the  point  B  lying  on  OA.  Hence, 
the  farther  R  (Fig.  63)  lies  from  X,  the  shorter  does  the  cor- 
responding radius  of  the  ray-surface  (Fig.  64)  become;  and 


132  OPTICALLY   BIAXIAL  CRYSTALS 

when  R  coincides  with  Z,  so  too  does  N  coincide  with  O, 
and  Or  with  OX.  That  is  to  say,  the  ray  transmitted  along 
the  direction  OX,  which  vibrates  parallel  to  OZ,  has  the  velocity 
proportional  to  i/f —  the  least  in  the  crystal.  Consequently, 
for  the  extraordinary  rays  vibrating  in  the  principal  section 
XZXZ,  when  the  transmission  direction  varies  from  OZ  to 
OX  the  velocity  varies  from  OA  to  OC  (Fig.  64) ;  these  two 

lengths  have  the  ratio  —  :  —  (=  ~f\  a).     If,  then,  we  imagine  the 

resulting  lengths  as  laid  off,  in  correspondence  to  the  symmetry 
of  the  index-surface,  in  all  the  four  quadrants  lying  between 
the  axes  XX  and  ZZ,  we  obtain  the  following  curves  as  the 
intersection  of  the  ray-surface  with  the  principal  section  XZXZ : 
for  the  extraordinary  ray  an  ellipse,  whose  axes  are  propor- 
tional to  the  largest  and  the  smallest  refractive  index;  tfor  the 
ordinary  ray  a  circle,  with  the  diameter  proportional  to  i/j(3.  In 
consequence  of  this  relation  between  the  diameters  of  the  two 
curves,  the  curves  must  intersect  each  other  four  times;  and 
therefore  two  directions  exist,  M 1M1  and  M2M2,  in  which  the 
transmission  velocity  of  the  two  rays  is  equal.  Sihce  in  these 
directions  one  length  is  radius  vector  at  once  for  both  skins  of 
the  ray-surface,  these  directions  have  been  named  bi-radials; 
they  are  known  besides  as  ray-axes,  or  as  secondary  optic  axes  in 
distinction  from  the  two  directions  mentioned  at  the  end  of  the 
last  section,  the  so-called  "  primary  optic  axes  ",  which  lie  in 
the  same  plane  and  likewise  present  certain  analogies  with  the 
optic  axis  of  uniaxial  crystals. 

Now  as  for  the  intersection  of  the  ray-surface  with  the 
second  principal  section,  FZFZ,  it  has  already  been  shown  that 
one  of  its  curves  is  a  circle,  with  the  radius  proportional  to  the 
greatest  light  velocity.  The  second  curve  has  a  variable  radius 
vector.  For  if  we  start  out  from  the  point  Y  of  the  index- 
surface,  there  corresponds  to  this  point  in  the  direction  OZ  an 
extraordinary  ray  with  the  vibration  direction  OF  and  therefore 
with  the  velocity  proportional  to  i//?  =  OB  (Fig.  65);  to  points 


RAY-SURFACE 


133 


lying  between  Y  and  Z  on  the  same  principal  section  of  the 
index-surface  there  corresponds  greater  length  of  the  normals 
(since,  of  all  the  normals  in  this  principal  section,  that  at  Y 
is  the  shortest)  and  therefore  less  velocity  of  the  rays  proper 
to  them;  at  Z,  finally,  the  normal  to  the  surface  is  longest,  so 
the  rays  vibrating  parallel  to  OZ  and  transmitted  along  the 
direction  OY  have  the  least  velocity  OC.  Of  the  light  rays 
propagated  in  the  principal  section  FZFZ,  therefore,  the  extraor- 


Fig.  65. 

dinary  advance  in  a  certain  time  as  far  as  an  ellipse  whose 
axes  are  to  each  other  as  the  intermediate  to  the  least  light 
velocity,  i.e.  as  the  smallest  refractive  index  to  the  intermediate; 
while  the  ordinary,  in  that  same  time,  have  been  propagated 
as  far  as  a  circle  with  the  radius  proportional  to  the  greatest 
light  velocity,  — proportional  therefore  to  the  reciprocal  of  the 
smallest  refractive  index.  So  the  circle  surrounds  the  ellipse 
without  touching  it. 

The  curves  corresponding  to  the  third  principal  section, 
XYXY,  are  shown  in  Fig.  66  on  the  next  page.  One  of  them 
is  a  circle  with  the  radius  OC;  it  contains  all  the  rays  that  cor- 


OPTICALLY   BIAXIAL   CRYSTALS 


respond  to  the  point  Z  on  the  index-surface,  these  rays  all  having 
the   vibration   direction  OZ  a$d  therefore  the  constant  trans- 


mission velocity  OC  proportional  to  - 


The  extraordinary  ray 


transmitted  along  the  same  direction  as  any  one  of  these  ordinary 
rays  always  has  a  greater  velocity,  since  the  point  corresponding 
to  it  on  the  index-surface  lies  in  the  principal  section  X YXY, 
wherefore  the  normal  erected  at  that  point  is  shorter  than  OZ 


(this being  altogether  the  longest  that  exists).  The  transmission 
velocity  of  the  extraordinary  rays  transmitted  in  this  principal  sec- 
tion reaches  its  maximum,  OA,  for  the  ray  vibrating  parallel  to 
OX  and  hence  having  the  direction  OF;  its  minimum,  OB,  for 
the  vibration  parallel  to  OF,  — i.e.  for  the  ray  direction  OX. 
Of  the  ray-surface  intersections,  accordingly,  the  curve  corre- 
sponding to  the  ordinary  rays  lies  wholly  within  the  ellipse 
whose  radii  vectores  represent  the  transmission  velocities  of  the 
extraordinary  rays. 

So  far  as  is  possible  without  a  model,  the  perspective  view  in 
Fig.  67  may  serve  to  give  a  more  connected  idea  of  the  form  of 


RAY-SURFACE 


135 


the  entire  ray-surface  derived  from  the  index-surface;  it  is  to  this 
surface,  therefore,  that  a  light-motion  of  a  definite  color,  begin- 
ning at  the  center,  has  been  transmitted  after  a  definite  time. 
In  the  figure  the  space  within  the  inner  skin  of  the  surface  is 
shaded,  while  that  included  between  the  two  skins  is  white. 

This  double  surface  was  derived  first  by  Fresnel,  although 
in  a  somewhat  different  way;  he  called  it  the  "  wave-surface  ", 
and  showed  that  through  it  the  optical  properties  of  the  biaxial 


crystals  could  be  completely  explained.     It  is  for  this  reason 
often  designated  also  as  "  Fresnel' s  surface  ".* 

As  follows  from  the  foregoing,  the  form  of  this  surface  is 
completely  determined  when  the  greatest,  the  intermediate,  and 
the  least  light  velocity,  with  their  respective  directions  in  the 
crystal,  are  known.  Supposing  these  data  to  have  been  de- 
termined (by  methods  to  be  discussed  in  the  next  section),  then 

*  Models  of  FresnePs  wave-surface  in  gypsum  can  be  obtained  from  M. 
Schilling  in  Leipzig;  in  brass  wire,  after  specifications  of  the  author,  from  Bohm 
and  Wiedemann  in  Munich.  (See  Appendix.)  Polished  wooden  models  of  the 
corresponding  index-surfaces,  which  can  be  taken  apart,  are  supplied  by  G.  J. 
Pabst  of  Nuremberg.  (See  ibid.) 


136 


OPTICALLY   BIAXIAL   CRYSTALS 


for  every  ray  entering  the  crystal  in  any  direction  we  may 
determine,  by  means  of  Huygens's  construction,  the  direction  in 
which  it  is  refracted,  just  as  was  done  on  page  95  et  seq.  for  the 
uniaxial  crystals.  If  we  carry  out  the  construction  for  a  ray 
whose  plane  of  incidence  coincides  with  one  of  the  three  prin- 
cipal sections,  e.g.  for  the  parallel  rays  DB  to  CO,  Fig.  68,  — 
whose  common  ray-front  is  OE  and  whose  common  plane  of  inci- 
dence is  parallel  to  FOZ,— then  at  the  instant  when  DB  enters 
the  crystal  the  ray-fronts  of  the  two  light-motions  arising  by 


Fig.  68. 

double  refraction  will  be  the  tangential  planes  from  B  to  the 
two  skins  of  the  ray-surface;  and  one  easily  sees  that  owing  to 
the  symmetry  of  the  latter  with  reference  to  the  principal  sec- 
tion YOZ  the  two  points  of  tangency,  o  and  e,  lie  in  the  same 
principal  section,  and  accordingly  also  that  although  the  two 
rays  Oo  and  Oe  are  indeed  deflected  they  do  not  leave  the  prin- 
cipal section  YOZ.  If,  however,  the  plane  of  incidence  is 
parallel  to  none  of  the  three  principal  sections,  then,  with  a  con- 
struction analogous  to  the  last,  the  ray-surface  intersection 
parallel  to  the  plane  of  incidence  will  divide  that  surface  into 
unequal  halves,  so  that  the  halves  lying  respectively  before  and 
behind  the  plane  of  the  figure  do  not  lie  symmetrically  with  ref- 


RAY-SURFACE 


erence  to  this  plane.  The  points  where  the  tangential  planes 
representing  the  refracted  ray-fronts  touch  the  two  skins  of  the 
ray-surface  will  then  lie  no  longer  in  the  plane  of  the  figure,  but 
before  or  behind  it.  Accordingly,  the  refracted  rays  are  both 
deflected  from  the  plane  of  incidence;  that  is  to  say,  neither  of 
them  any  longer  follows  the  law  of  refraction  for  ordinary  light: 
both  are  extraordinary.  So  an  ordinary  ray  (in  addition  to  an 
extraordinary)  is  obtained  only  when  the  plane  of  incidence  is 
parallel  to  one  of  the  three  principal  sec- 
tions. Since  by  means  of  methods  based 
on  the  law  of  refraction  for  ordinary  light 
we  can  in  general  determine  the  light  ve- 
locity only  of  such  rays  as  follow  that  law, 
i.e.  of  ordinary  rays,  the  property  of,  bi- 
axial crystals  just  now  set  forth  suggests 
at  the  same  time  the  most  convenient 
methods  for  determining  the  light  velocity 
in  one  of  these  crystals.  (For  particu- 
lars of  these  methods  see  next  section.) 

The  foregoing  considerations  and  the 
figures  62-66  have  been  based  on  a  special 
example,  in  which  the  value  of  the  inter- 
mediate refractive  index  /?  lies  closer  to 
that  of  the  smallest  a  than  to  that  of  the 
largest  7-.  The  form  of  the  index-surface 
in  Fig.  69  therefore  approximates  — and 
if  the  difference  between  /?  and  a  were  still  less  this  would  be 
the  case  in  a  still  higher  degree  —  to  that  of  a  spheroid  hav- 
ing OZ  as  rotation  axis;  and  it  passes  over  into  such  a  spheroid 
when  a  =  /?,  — i.e.  when  OX  =  OY.  Since  this  extreme  case 
corresponds  to  a  positive  optically  uniaxial  crystal,  those, biaxial 
crystals  also  are  designated  as  positive  in  which  /?  lies  closer 
to  a  than  to  f.  But  the  less  the  lengths  OX  and  O  Y  differ  from 
each  other,  the  smaller  is  the  angle  included  between  OX  and  the 
radius  vector  OCX,  equal  in  length  to  OF;  and  consequently  the 


I38 


OPTICALLY   BIAXIAL   CRYSTALS 


smaller  also  is  the  angle  that  the  two  optic  axes  —  standing,  as 
they  do,  perpendicular  respectively  to  the  circular  sections  C1Cl 
and  C2C2 — form  with  OZ.  Optically  positive  biaxial  crystals, 
therefore,  are  those  in  which  the  vibration  direction  of  the  rays 
transmitted  the  most  slowly  (those  with  the  largest  refractive  in- 
dex) is  the  so-called  acute  bisectrix  (first  mean  line)  of  the  optic 
axes;  i.e.  bisects  their  acute  angle. 

Conversely,  when  p  is  the  closer  in  value  not  to  a  but  to  7-, 

i.e.  OY  but  little  different 
from  OZ,  the  form  of  the 
index-surface  approximates 
to  a  spheroid  with  the  rota- 
tion axis  OX ;  in  other  words, 
to  the  index-surface  of  a 
negatively  uniaxial  crystal. 
The  value  /?  for  a  radius 
•*  vector  of  the  index-surface 
between  OX  and  OZ  is  then 
attained  only  near  the  latter; 
so  the  two  circular  sections 
form  an  acute  angle  with  OZ, 
and  the  optic  axes  normal  to 
them  an  obtuse  angle.  Such 
crystals  are  said  to  be  nega- 


Fig.  70. 


tively  biaxial,  and  are  char- 
acterized by  the  fact  that  the 
vibration  direction  of  the  rays  transmitted  the  most  slowly  is  the 
obtuse  bisectrix*  (second  mean  line)  of  the  optic  axes;  i.e.  bisects 
their  obtuse  angle.  In  correspondence  tojthis,  in  a  crystal  of  the 
latter  sort  the  bi-radials,  MlMl  and  M2M2,  lie  as  represented  in 
Fig.  70. 

Since  the  quantities  a,  /?,  f  are  different  for  different  colors, 
there  are  biaxial  crystals,  also,  that  for  certain  colors  are  posi- 

*[The  term  "  bisectrix"  when  used  alone,  without  special  qualification,  refers 
to  the  acute  bisectrix.] 


RAY-SURFACE 


139 


tive  and  for  others  negative,  the  angle  between  the  axes  amount- 
ing, for  example,  for  one  color  to  89°  and  for  another  to  91°; 
there  is  then  a  color  for  which  the  crystal  is  exactly  interme- 
diate between  positive  and  negative,  thus  being  neither  the  one 
nor  the  other. 

Now  it  yet  remains  to  deduce  from  the  ray-surface  the  rela- 
tions in  which  the  two  so-called  primary  optic  axes  stand  to  the 


secondary,  the  ray-axes,  and  to  learn  in  what  way  these  directions 
depend  on  the  values  of  the  three  principal  refractive  indices. 

To  this  end  let  us  first  consider  somewhat  more  closely  the 
properties  of  the  rays  transmitted  parallel  to  the  two  kinds  of 
optic  axes. 

As  the  form  of  the  ray-surface  teaches,  the  two  rays,  arising 
in  any  direction  are  in  general  transmitted  with  unequal  veloc- 
ity; only  in  the  direction  of  a  so-called  bi-radial,  or  ray-axis,  as 
OM  in  Fig.  71,  are  these  two  rays  transmitted  equally  fast. 


140 


OPTICALLY   BIAXIAL   CRYSTALS 


Such  a  direction  differs  essentially,  however,  from  the  optic 
axis  of  a  uniaxial  crystal,  in  that  although  the  two  rays  in  ques- 
tion have  indeed  the  same  transmission  velocity,  their  respec- 
tive ray-fronts  have  not  the  same  direction.  For  the  front  of  the 
ordinary  ray  —  the  plane  perpendicular  to  the  principal  section 
XZ  through  the  tangent,  kk,  to  the  circular  intersection  of  the 
ray-surface  at  M  —  stands  perpendicular  to  OM ,  while  on  the 


Fig.  71. 

other  hand  the  front  of  the  extraordinary  ray  of  the  same  trans- 
mission direction  stands  oblique  to  OM,  it  being  the  plane  per- 
pendicular to  the  principal  section  XZ  through  kfkf,  the  tangent 
at  M  to  the  elliptical  intersection  of  the  ray-surface.  The  dis- 
tance of  these  two  ray-fronts  from  the  point  whence  the  light- 
motion  proceeds,  i.e.  the  length  of  the  normals  let  fall  from 
the  center  O  upon  the  two  planes,  is  consequently  different;  in 
other  words,  the  transmission  velocities  of  these  two  wave- 
motions,  as  measured  along  their  front-normals,  are  unequal. 


RAY-SURFACE  141 

But  the  velocities  so  measured  come  into  consideration  only  in 
the  case  of  optically  isotropic  media,  where  ray  and  front- 
normal  are  identical;  and  hence  it  follows  that  the  two  rays 
passing  through  the  crystal  in  the  direction  OM  with  equal 
velocity  are,  in  correspondence  to  the  difference  in  their  ray- 
fronts,  on  emerging  into  the  air  differently,  i.e.  doubly,  refracted. 
Now  these  two  fronts  are  not  the  only  tangential  planes  to  the 
ray-surface  at  the  point  M\  for,  since  at  this  place  there  is  a 
conical  depression  in  the  outer  skin,  an  infinity  of  planes  can 
be  passed  through  the  point  M  tangent  to  this  skin.  The  nor- 
mals to  these  planes,  i.e.  the  rays  in  the  air  proper  to  the  several 
fronts,  form  an  acute  cone.  Hence,  if  such  a  cone  of  converg- 
ing rays  is  made  to  fall  on  a  plane-parallel  biaxial  crystal  plate 
cut  perpendicular  to  a  ray-axis  OM,  these  rays,  within  the 
crystal,  are  all  transmitted  along  the  direction  OM  and  with 
equal  vejpcity,  and  on  emerging  they  are  refracted  into  a  similar 
cone  (exterior  conical  refraction). 

While  with  the  uniaxial  crystals  there  exists  at  both  extremi- 
ties of  the  axis  a  common  tangential  plane  to  the  two  skins,  this 
is  not  the  case  at  the  four  umbilical  points  of  the  biaxial  ray- 
surface  (e.g.  at  'M)',  at  which  points  the  two  skins  penetrate 
each  other.  But  on  the  other  hand,  over  each  of  these  four 
depressions  in  the  outer  skin  there  is  a  common  tangential 
plane  to  the  two  skins;  that  over  M,  for  example,  is  the 
plane  passed  perpendicular  to  the  principal  section  XZ  through 
the  straight  line  ft,  which  is  tangent  to  the  ellipse  at  U  and  to 
the  circle  at  U'.  This  plane  touches  the  ray-surface  around  the 
point  M  along  a  closed  curve,  to  which  the  points  U  and  U' 
belong;  and  from  the  geometrical  properties  of  the  ray-surface 
it  follows  that  this  curve  is  a  circle.  Hence,  if  we  imagine  radii 
vectores  as  drawn  to  all  points  of  this  circle  (visible  in  Fig.  71 
are  the  two  radii,  OU  and  OU',  that  lie  in  the  plane  of  the 
figure)  we  obtain  a  cone  of  rays  all  having  one  common  ray- 
front;  namely,  the  plane  passed  through  tt  perpendicular  to  the 
principal  section  XZ.  Consequently,  also  the  normal  to  this 


142  OPTICALLY   BIAXIAL   CRYSTALS 

plane,  the  front-normal  of  these  rays,  is  common  to  them  all; 
for  example,  the  front-normal  Uu  of  the  ray  OU  is  parallel  to 
the  front-normal  U'u'  of  the  ray  OZ7';  and  so  likewise  are  the 
front-normals  Vv  and  FV  of  the  rays  OV  and  OV  parallel  to 
each  other.  Now  these  two  directions  Uu  and  Vv  (or  OU'  i 
and  OV)  are  — as  follows  from  the  equation  of  the  index-/ 
surface- — nothing  else  than  the  so-called  primary  optic  axes;j 
i.e.  the  normals  to  the  circular  sections  of  the  index-surface  from 
which  the  ray-surface*  is  derived.  So  the  behavior  of  all  rays 
transmitted  parallel  to  a  primary  optic  axis,  whose  ray-front 
therefore  is  the  plane  of  the  circular  section,  is  analogous  to  that 
of  the  rays  transmitted  parallel  to  the  optic  axis  of  a  uniaxial 
crystal,  inasmuch  as  their  vibrations  in  this  plane  can  have  any 
azimuth.  The  primary  optic  axes  accordingly  exhibit  a  still 
closer  analogy  with  the  optic  axis  of  uniaxial  crystals  than  do 
the  ray-axes,  and  will  therefore  in  the  following  be  designated 
briefly  as  optic  axes.*  If  parallel  to  a  circular  section  of  the 
index-surface,  i.e.  perpendicular  to  an  optic  axis,  we  imagine  a 
plane  face  as  ground  on  the  crystal,  and  cause  parallel  light 
rays  to  fall  perpendicularly  on  this  face,  then  all  the  rays  that 
belong  to  a  circular  cylinder  will,  after  their  entry  into  the 
crystal,  form  a  cone  with  the  conical  angle  UOU'  (interior  coni- 
cal refraction).  Conversely,  all  the  diverging  rays  transmitted 
within  a  biaxial  crystal  along  the  convex  surface  of  this  cone 
will,  on  emerging  through  the  plane  face  in  question,  become 
parallel  and  be  transmitted  in  the  air  as  a  ray  cylinder. 

Now  the  aperture  of  this  cone  of  interior  conical  refraction  is 
in  reality  very  much  smaller  than  represented  in  Fig.  71  (seldom 
over  2°),  because  the  difference  among  the  principal  refractive 
indices,  and  therefore  the  diversity  in  the  transmission  velocities 
of  rays  vibrating  in  different  directions,  is  never  so  great  as  was, 

*  Since  OU'  (Fig.  71)  is  perpendicular  to  //,  it  is  parallel  to  such  an  optic  axis, 
and  the  same  line  OU'  is  the  normal  both  for  the  ray-front  at  the  point  U'  and  for 
the  ray-front  at  the  point  U.  Fletcher  proposed  for  the  two  optic  axes  the  name 
bi-normals,  for  the  reason  that  here  it  is  a  matter  of  directions  in  which  one  line 
is  in  a  double  sense  the  normal  to  a  ray-front. 


RAY-SURFACE 


for   the    sake   of   distinctness,    assumed  in  Fig's  62-71.     The 
same  applies  also  to  the  cone  of  exterior  conical  refraction. 

Let  the  angle  between  the  two  optic  axes  (optic  axial  angle) , 
i.e.  the  angle  U'OV  in  Fig.  71,  be  represented  by  2V:  let  V 
(=  U'OZ  in  Fig.  71)  thus  be  the  angle  included  between  either 
optic  axis  and  the  vibration  direction  of  the  rays  transmitted  in 
the  crystal  the  most  slowly.  From  the  equation  of  the  index- 
surface  there  follows  for  this  angle 


tang  V  = 


If  8  so  lies  between  a  and  7-  that —  =  - —  —     then   is 

or     ft1      /92      f 

tang  V  =  i,  and  V  therefore  =  45°.     If  /?  is  more  nearly  equal  to 


a  than  to  7%  and  in  consequence  the  numerator  of  the  fraction  under 
the  square-root  sign  less  than  the  denominator,  then  V  is  less  than 
45°;  that  is,  we  have  to  do  with  a  positively  biaxial  crystal,  in 


146  OPTICALLY    BIAXIAL    CRYSTALS 

perpendicular  to  this  plane,  experiences  the  minimum  deviation 
when  it  passes  in  the  prism  from  a  to  b;  it  then  enters  the  prism 
in  the  direction  oa  and  emerges  in  the  direction  bo'.  If  we  so 
turn  the  prism,  or  if  we  shift  the  source  of  light  in  such  a  way, 
that  the  extraordinary  ray  on  its  part  suffers  the  minimum 
deviation  (which  in  this  case  is  less  than  with  the  ordinary  ray), 
it  enters  the  prism  in  the  direction  ea  and  emerges  along  be' , 
thus  passing  through  the  prism  in  the  same  direction  ab  as  did 
previously  the  ordinary  ray;  i.e.  in  the  direction  XX.  But  its 
vibration  direction  is  then  obviously  parallel  to  OF,  and  its 
velocity  therefore  the  intermediate;  so  that  if  we  measure  the 
deviation  for  this  position  and  calculate  the  proper  refractive 
index,  this  latter  has  exactly  the  value  /?.  Since  the  ordinary  ray 
gives  us  7-  (because  it  has  the  vibration  direction  OZ),  then  with 
the  aid  of  this  one  prism  we  obtain  two  of  the  principal  refractive 
indices.  In  the  same  way  the  investigation  of  a  second  prism, 
whose  faces  are  parallel  to  OX  and  equally  inclined  toXOZ, 
supplies  «  and  7-;  a  third  prism,  symmetrical  to  XOF,  with  its 
edge  parallel  to  OF,  gives  us  a  and  /?.  Accordingly,  only  two 
such  prisms  are  necessary,  to  determine  all  three  principal  re- 
fractive indices. 

The  same  is  the  case  moreover  when  the  prisms  are  cut  in 
still  another  direction;  namely,  when  one  of  the  lateral  faces 
coincides  with  a  principal  optic  section  of  the  crystal.  Let 
PP'P'  (Fig.  74)  be  such  a  prism  with  the  refracting  angle  w, 
whose  edge  is  parallel  to  OZ  and  whose  left  face  is  parallel  to 
the  principal  section  FOZ.  If  light  rays  /,  I',  I",  etc.,  are  caused 
to  fall  perpendicularly  on  this  face  and  therefore  parallel  to  XX, 
the  two  ray-fronts  transmitted  in  the  crystal  are  the  tangential 
planes,  U  and  t't'9  to  the  ray-surfaces  to  be  constructed,  all  of 
equal  dimensions,  from  each  point  of  entrance,  —  so  constructed 
because  the  incident  plane  front  strikes  all  the  points  of  entrance 
at  the  same  time.  From  the  construction,  and  from  the  sym- 
metry of  the  wave-surface  with  reference  to  the  principal  sections, 
it  follows  directly  that  the  two  tangential  planes  are  exactly  per- 


DETERMINATION  OF   PRINCIPAL   REFRACTIVE   INDICES      147 

pendicular  to  the  plane  of  the  figure  and  exactly  parallel  to  each 
other,  as  also  to  the  face  of  entrance  of  the  light.  So  the  two 
rays  experience  no  deviation  whatever,  but  are  both  transmitted 
parallel  to  OX,  as  in  the  prism  of  Fig.  73;  the  one  therefore  with 


x 1 


Fig.  74- 


the  least  velocity,  the  other  with  the  intermediate;  and  accord- 
ingly, on  their  exit  at  x  they  are  differently  refracted,  the  one  to 
o,  the  other  to  e.  If  we  determine  the  deviation  of  each  ray  from 
its  original  direction  and  denote  these  deviations  by  o>  and  e  re- 
spectively, the  former  is  the  angle  Xxo,  the  latter  Xxe.  As  is 
manifest  from  Fig.  74  directly,  the  refractive  indices  of  the  two 
rays  are  then 

sin(&>  +  w} 

y    =     :: i 

sin  w 

~  _  sin  (s.  +  w) 
sin  w 

By  means  of  such  a  prism,  therefore,  two  principal  refractive 
indices  may  be  determined;  and  thus,  by  means  of  two  differ- 


148  OPTICALLY   BIAXIAL   CRYSTALS 

ent  prisms,  each  of  which  has  one  side  parallel  to  a  different 
principal  section,  we  can  determine  all  three. 

REMARK.  —  When  the  optic  axial  angle  is  known,  the  measurement  or  calcu- 
lation of  the  three  principal  refractive  indices  may  be  accomplished  also  with  a 
prism  of  different  orientation.  (See,  for  example,  Hutchinson,  Zeitschr.  f.  Kryst. 
1901,  34,  347-) 

The  determination  of  the  three  principal  refractive  indices 
can  be  carried  out  to  better  advantage  with  the  aid  of  the  total- 
reflectometer,  because  with  this  method  only  one  single  plate  of 
the  crystal  is  requisite,  and  this  plate  need  fill  only  one  con- 
dition; namely,  that  it  be  parallel  to  one  of  the  three  principal 
vibration  directions.  If  such  a  plate  is  brought  into  the  instru- 
ment (Fig.  22,  p.  40)  and  turned  so  that  this  direction,  lying  in 
the  plate,  coincides  with  the  plane  of  incidence  of  the  light,  then, 
when  the  plate  has  been  rotated  to  the  angle  of  total  reflection, 
the  light  is  transmitted  parallel  to  the  principal  vibration  direc- 
tion in  question  and  is  split  up  therefore  into  two  rays  that 
vibrate  parallel  to  the  two  other  principal  vibration  directions; 
accordingly  the  adjusting  of  the  double  boundaries  of  total  re- 
flection supplies  two  principal  refractive  indices.  Now  if  we 
rotate  the  plate  90°  in  its  own  plane  and  fasten  it  in  this  position 
to  the  rotation  axis  of  the  instrument  (the  latter  is  then  parallel 
to  the  principal  vibration  direction  falling  in  the  plane  of  the 
plate),  then  on  the  total  reflection  the  transmission  direction  of 
the  light,  although  in  general  it  will  be  parallel  to  no  principal 
vibration  direction,  will  nevertheless,  since  it  is  perpendicular  to 
one  of  these,  lie  in  a  principal  section;  therefore  one  of  the  two 
rays,  the  ordinary,  arising  by  double  refraction  vibrates  parallel 
to  the  principal  vibration  direction  that  in  the  first  case  was 
transmission  direction,  and  consequently  the  adjustment  of  the 
critical  angle  peculiar  to  this  ray  gives  the  third  principal  re- 
fractive index.  In  order  to  distinguish  the  two  boundaries  from 
each  other,  one  brings  before  the  telescope  of  the  total-reflec- 
tometer  a  Nicol  prism  with  its  principal  section  vertical.  This 
transmits  only  the  ray  vibrating  vertically,  and  therefore  only 


DETERMINATION  OF  PRINCIPAL  REFRACTIVE  INDICES      149 

that  boundary  of  total  reflection  appears  in  the  field,  that  sup- 
plies the  third  refractive  index.*  It  is  easy  to  see  that,  with  the 
use  of  a  plate  parallel  to  a  principal  optic  section,  if  the  plate 
is  so  oriented  that  first  the  one  and  then  the  other  of  the  two 
principal  vibration  directions  lying  in  the  plane  of  the  plate 
becomes  transmission  direction  of  the  light  on  the  total  reflec- 
tion, we  obtain  each  time  two  principal  refractive  indices;  with 
such  a  plate,  therefore,  we  may  not  only  determine  all  three 
principal  refractive  indices,  but  one  of  them  even  twice.  Of 
special  interest  are  the  phenomena  that  a  crystal  plate  cut  paral- 
lel to  the  optic  axial  plane  exhibits  in  the  total-reflectometer, 
when  the  plate  is  made  rotatable  in  its  own  plane.  Starting  out 
with  the  plate  so  oriented  that  the  plane  of  incidence  is  parallel 
to  a  principal  vibration  direction  of  the  plate,  one  then  sees  two 
boundaries  of  total  reflection.  Now  if  the  plate  is  rotated  in  its 
own  plane  these  two  boundaries  approach  each  other,  because 
the  two  rays  transmitted  in  the  boundary-layer  of  the  plate 
always  have  less  different  velocity.  And  when  the  plate  has  been 
rotated  until  an  optic  axis  becomes  transmission  direction,  the 
two  boundaries  of  total  reflection  coincide;  so  that  in  the  field 
of  view,  besides  the  vertical  boundary,  corresponding  to  the  ordi- 
nary ray,  there  appears  the  second  boundary  of  total  reflection, 
crossing  the  former  at  the  center  of  the  field  at  an  acute  angle. 
Provided  this  angle  is  not  too  sharp,  that  is  to  say,  provided  the 
transmission  velocity  of  the  extraordinary  ray  varies  rapidly 
with  the  direction  (i.e.  the  crystal  has  high  birefringence),  a 
rotation  of  the  plate  through  360°  in  its  own  plane  accordingly 
brings  into  view,  little  by  little,  the  whole  intersection  of  the 
ray-surface  with  the  principal  section  XZ.  (See  Fig.  64,  in 
which  the  ratio  of  the  greatest  to  the  least  light  velocity  is  of 
course  much  exaggerated;  see  also  Fig.  72,  p.  143;  and  cf. 
Fig.  75,  p.  150,  with  the  explanation.)  With  this  method  as  a 

*  It  is  hardly  necessary,  after  the  previous  expositions,  to  remark  that  in  every 
determination  of  the  refractive  indices  of  a  doubly  refracting  body  the  vibration 
directions  of  the  several  rays  must  be  determined,  by  means  of  a  nicol. 


150  OPTICALLY   BIAXIAL   CRYSTALS 

basis  W.  Kohlrausch  measured  the  transmission  velocity  of  light 
in  numerous  directions  within  the  three  principal  optic  sections 
of  tartaric  acid  (cf.  remark  on  p.  144),  and  found  it  wholly  in 
keeping  with  Fresnel's  surface.* 

Finally,  the  three  principal  refractive  indices  can  be  de- 
termined also  through  total  reflection  from  a  biaxial  crystal  plate 
of  any  orientation,  by  measuring  the  maxima  and  minima  of  the 
two  curves  that  such  a  plate  exhibits  [in  Abbe's  refractometer. 
(See  pp.  42-44-)] 

Should  we  here  employ  a  cylindrical  plate  (see  Fig.  24  /.  c.)  and  illuminate 
the  cylindrical  surface  from  all  sides,  then  on  a  white  screen  placed  beneath 
the  glass  sphere  we  should  see  the  closed  boundary-curves  of  total  reflection. f 
While  in  the  case  of  a  uniaxial  crystal  plate  parallel  to  the  axis  these  curves 
have  a  form  similar  to  that  of  Fig.  50  (p.  94)  or  of  Fig.  56  (p.  102),  and  in 

the  case  of  a  biaxial  crystal  parallel  to  the  axial 
plane  a  form  as  in  Fig.  72  (p.  143),  with  a  biaxial 
plate  oriented  at  random  there  appear  in  general 
two  curves  as  in  Fig.  75,  where  the  radius  vector  of 
each  curve  has  a  maximum  and  a  minimum  value. 
Now  with  Abbe's  crystal  refractometer  the  four 
values  of  the  light  velocity  corresponding  to  these 
maxima  and  minima  can  be  very  accurately  deter- 
mined, by  measuring,  during  a  rotation  of  360°, 
the  refractive  indices  in  a  considerable  number 
of  azimuths;  and,  since  the  rotation  angles  may 
be  read  off  on  the  horizontal  graduated  circle,  we 
can  also  ascertain  the  orientation  of  the  radii  in  question.  Hence  one  of 
these  radii,  «,  corresponds  to  the  absolute  minimum,  and  therefore  to  the 
smallest  refractive  index  of  the  crystal;  the  greatest,  r,  to  the  absolute  maxi- 
mum, and  accordingly  to  the  largest  refractive  index;  while  to  the  inter- 
mediate refractive  index  of  the  crystal  there  corresponds  either  the  maximum  0 
or  the  minimum  /?'4 

*  For  the  details  of  such  measurements  as  the  foregoing  and  that  next  follow- 
ing cf.  footnote  t,  P-  3°- 

f  Apparatus  for  projecting  and  photographing  the  boundary-curves  have 
been  constructed  by  Leiss  (see  Zeitschr.  f.  Kryst.  1898,  30,  357)  and  are  supplied 
by  the  firm  of  Fuess.  (See  also  Die  op.  Instrum.,  49~52-) 

J  For  particulars  see  Viola,  Zeitschr.  f.  Kryst.  1899,  31,  40  et  seq.;  1900,  32. 
311  et  seq.,  and  33,  30  et  seq.;  1902,  36,  245  et  seq.;  1904,  39,  179  et  seq. 


DETERMINATION   OF   PRINCIPAL  REFRACTIVE   INDICES      151 

Supposing  a,  /?,  and  7-  to  have  been  determined  by  one  of 
the  methods  just  discussed,  then  from  them  the  axial  angle 
2  V  may  be  calculated,  according  to  the  formula  given  on  page 
143.  On  the  other  hand,  if  V  itself  has  been  measured  (by  a 
method  to  be  discussed  farther  on),  and,  besides  this  quantity, 
only  two  of  the  three  principal  refractive  indices  (when,  for 
example,  the  development  of  the  crystal  permits  the  prepara- 
tion of  prisms  only  in  one  direction),  then  by  means  of  the 
same  formula  we  may  calculate  the  third  principal  refractive 
index. 

The  numbers  by  which  the  optical  properties  of  a  crystal  are 
completely  denned  we  call  its  optical  constants;  with  an  optic- 
ally biaxial  crystal  these  are  the  cry stallo graphic  orientations  (f 
the  three  principal  vibration  directions  and  the  values  of  the  three 
principal  refractive  indices.  These  quantities  refer  in  general 
only  to  one  definite  color,  and  therefore  all  that  has  been  said 
up  to  the  present  concerning  the  index-surface  and  the  ray- 
surface  applies  only  to  light  of  one  definite  vibration  period. 
If  for  determining  the  three  principal  refractive  indices  we  em- 
ploy light  of  another  color,  we  find  other  values  for  them;  and 
what  is  more,  with  each  index  the  variation  with  the  vibration 
period  is  different;  that  is  to  say,  for  a,  /?,  and  7-  the  constants  of 
Cauchy's  dispersion  formula  (see  p.  46)  have  different  values. 
In  consequence  of  this,  for  a  different  color  the  values  of  the 
greatest,  the  intermediate,  and  the  least  light  velocity  stand  in 
different  ratios;  in  other  words,  the  ray-surface  (like  the  index- 
surface)  for  the  latter  color  has  a  different  shape,  and  the  optic 
axes  form  a  different  angle  with  each  other;  this  angle  increas- 
ing with  one  substance  with  the  wave  length  of  the  light,  with 
another  diminishing.  Therefore  the  refractive  indices  a,  /?,  7- 
must  always  be  measured  for  several  colors,  when  the  optical 
properties  of  a  crystal  are  to  be  determined.  As  for  the  positions 
of  the  three  principal  vibration  directions  in  the  crystal,  these 
for  different  colors  may  be  the  same  or  different.  They  are 
determined  with  the  aid  of  the  interference  phenomena  that 


152  OPTICALLY   BIAXIAL   CRYSTALS 

biaxial  crystal  plates  exhibit  in  certain  directions,  wherefore 
these  phenomena  must  now  be  discussed  first;  and  here,  too, 
the  consideration  shall  always  be  begun  with  the  more  simple 
case  —  that  of  monochromatic  light. 

INTERFERENCE  PHENOMENA  OF  BIAXIAL  CRYSTALS  IN 
PARALLEL  POLARIZED  LIGHT 

Light  rays  of  a  definite  color  falling  perpendicularly  on  a 
plate  cut  in  any  direction  from  a  biaxial  crystal  are  in  general 
each  split  up  into  two  (extraordinary,  according  to  p.  137)  rays; 
these  rays,  after  they  emerge  from  the  crystal  plate,  vibrate  at 
right  angles  to  each  other  and  have  a  difference  of  path,  in 
virtue  whereof,  between  crossed  nicols,  interference  takes  place, 
exactly  as  with  the  uniaxial  crystals.  (See  p.  107  et  seq.)  The 
incident  rays  have  in  this  case  a  common  ray-front,  which  is 
parallel  to  the  plane  of  the  crystal  plate.  Now  within  the  crystal 
there  belong  to  this,  as  to  every  other,  ray-front  two  light  rays  of 
different  transmission  direction;  for  if  two  parallel  planes  are  tan- 
gent to  the  two  skins,  respectively,  of  the  ray-surface  of  a  biaxial 
crystal,  the  two  points  of  tangency  do  not  lie  on  the  same  radius 
vector.  Tr^transmission  and  vibration  directions  of  the  two  rays 
having  parallel  ray-fronts,  i.e.  the  same  front-normal,  ar«  obtained 
in  a  more  simple  way  from  the  index-surface,  as  follows;  Through 
the  center  of  the  'index-surface  a  plane  is  passed  parallel  to  the 
^nt  of  the  incident  rays,  parallel  therefore  to  the  cryjtal  plate; 
in  general  this  plane  intersects^  the  index-surface  in  an  ellipse. 
(Fig.  76.)  The  extremities,  R  and  T1,  of  the  major  and  the 
minor  axis  of  this  ellipse  are  obviously  those  points  of  the  ellipse 
that  stand  respectively  at  the  greatest  and  the  least  distance, 
measured  along  the  normals  to  the  ellipsoid  (RN  and  TN'  in 
this  case),  from  the  rays  proper  to  the  points;  and  the  transmis- 
sion direction  of  these  rays  is  obtained  if  from  O  we  let  fall  the  nor- 
mals to  RN.znd  TN'  (ON  and  ON'  respectively).  Thus  RN 
and  TN'  are  the  vibration  directions  of  the*  vibrations  transmitted 
the  slowest  (inversely  proportional  to  the  length  RN)  and  the 


INTERFERENCE    PHENOMENA   IN    PARALLEL   LIGHT       153 

fastest  proportional  to  (i/7W)  of  all  the  vibrations  that  corre- 
spond to  the  original  wave-plane  ORT-,  i.e.  to  the  plane  of  the 
crystal  plate.  So  any  vibration  taking  place  in  this  plane  is  on 
entering  the  crystal  plate  always  resolved  into  two  vibrations  RN 
and  TN' ,  which  are  transmitted  along  the  directions  Or  and  Ot 
respectively.  In  Fig.  76  the  vibration  planes  of  the  two  rays  Or 
and  Ot  are  hatched  parallel  to  the  vibration  directions  of  these 
rays.  These  two  vibration  directions,  RN  and  TN',  are  not  per- 
pendicular to  each  other;  the  two  vibration  planes,  however,  inter- 
sect in  O/at  right  angles.  Hence,  since  the  plane  face  through 


Fig.  76. 

which  the  light  emerges  from  the  crystal  plate  is  perpendicular 
to  Of,  and  since  on  emerging  the  two  rays  are  so  refracted  that 
they  again,  as  before  entering,  assume  the  direction  parallel  to 
Of  (because  in  the  air  front-normal  and  ray  are  identical),  then 
after  the  exit  RO  and  TO  are  the  two  vibration  directions.  In 
other  words,  there  are  then  transmitted  in  the  air  two  mutu- 
ally perpendicular  vibrations;  the  directions,  RO  and  TO,  of 
these  vibrations  are,  called  the  vibration  directions  of  the  crys- 
tal plate  in  question.  "These  directions  are  defined,  for  every  * 
plane  of  a  biaxial  crystal,  by  the  form  of  the  index-surface; 
and  conversely,  with  a  crystal  that  is  optically  known  their 
orientation  may  be  employed  for  determining  the  orientation  of 


154  OPTICALLY    BIAXIAL   CRYSTALS 

the  crystal  plate.  They  can  also  be  found  by  the  following 
construction:  - 

Through  the  normal  to  the  crystal  plate  (i.e.  the  front-normal 
of  the  incident  rays)  and  one  of  the  two  optic  axes  one  passes  a 
plane,  and  a  second  plane  is  passed  through  the  same  normal 
and  the  other  optic  axis.  These  two  planes  intersect  the  plane 
of  the  crystal  plate  in  two  straight  lines,  which  are  obviously 
normal  respectively  to  the  intersections  of  the  latter  plane  with 
the  planes  of  the  two  circular  sections.  Consequently  the  two 
diameters  of  the  ellipse  (Fig.  76)  that  are  parallel  to  the  lines 
in  question  are  of  equal  length;  and  therefore  the  bisectors 
of  the  two  angles  formed  by  these  two  lines  are  the  axes  of 
the  ellipse, —  i.e.  the  desired  vibration  directions  of  the  rays 
emerging  from  the  crystal  plate.* 

When  the  front-normal  Of  (Fig.  76)  is  parallel  to  an  optic 
axis,  the  intersection  RTRT  of  the  index-surface,  according  to  a 
previous  page,  is  a  circle. .  (See  p.  128.)  So  in  this  case  the  axes 
RR  and  TT  are  indeterminate;  that  is  to  say,  to  every  point 
lying  on  the  circumference  of  one  of  the  two  circular  sections  of 
the  index-surface  there  belongs  one  ray.  The  rays  belonging  to 

*  To  find  these  directions  graphically  it  is  best  to  make  use  of  a  model,  such 
as  that  supplied  by  Bohm  and  Wiedemann,  in  Munich.  (See  Appendix.)  This 
model  consists  of  a  horizontal  slate  tablet  representing  the  plane  of  the  crystal 
plate,  and  at  the  center  of  the  tablet  the  normal  is  represented  by  a  metal  bar. 
Rotatable  in  all  directions  about  the  foot  of  the  latter,  and  movable  at  will  with 
reference  to  each  other,  are  fitted  two  metal  bars  which  represent  the  two  optic 
axes.  Suppose,  then,  it  be  desired  to  determine  the  two  vibration  directions  for  a 
plate  cut  from  a  crystal  of  known  axial  angle  2V  in  a  direction  that  is  given  by 
the  inclination  of  the  plate  to  the  two  optic  axes.  With  the  aid  of  a  protractor 
(supplied  with  the  model)  one  sets  the  two  axes  at  the  correct  angle  zV  to  each 
other,  and  each,  with  reference  to  the  horizontal  plane,  in  the  direction  that  cor- 
responds to  the  orientation  of  the  plate;  one  then  lays  the  protractor  against  the 
normal  and  one  of  the  axes,  with  its  edge  resting  on  the  slate,  and  draws  the  cor- 
responding line  on  the  latter.  In  the  same  way  the  trace  on  the  slate  is  found  of 
the  plane  passed  through  the  normal  and  the  second  optic  axis;  and  it  is  then 
further  necessary  only  to  bisect  the  angle  these  two  traces  form  with  each  other. 
If  the  outline  of  the  crystal  in  the  plane  in  question  has  been  sketched  on  the  plate, 
then  with  the  protractor  one  can  now  read  directly  the  angles  the  vibration  direc- 
tions form  with  the  edges  of  the  plate. 


INTERFERENCE    PHENOMENA   IN   PARALLEL   LIGHT       155 

all  the  points  of  such  a  circle,  which  form  the  "  bi-normal 
cone",  or  "cone  of  interior  conical  refraction"  (see  p.  142), 
all  have  the  same  direction  and  position  of  their  ray-front,  this 
for  each  being  parallel  to  the 
circular  section;  their  several 
vibration  directions  therefore,  the 
normals  to  the  index-surface  at 
all  points  of  the  circular  section, 
can  have  any  azimuth.  As  may 
be  seen  from  Fig.  71,  p.  140,  one  of 
these  rays  coincides  with  the  optic 
axis  itself;  and  this  ray,  OM  in 
Fig.  77,  corresponds  to  the  points  Y 
and  Y  on  the  index-surface, —  the 
only  points  of  the  circular  section  lg'  77' 

CYCYoi  which  it  can  be  said  that  the  normal  to  this  surface  at 
the  point  in  question  passes  through  the  center  (because  YY  is 
one  of  its  axes,  and  because,  as  follows  from  Fig.  62,  no  other 
axis  of  the  index-surface  falls  in  the  circular  section).  There- 
fore the  vibration  direction  of  the  ray  OM  is  O  Y,  and  its  veloc- 
ity is  proportional  to  — —  =  — ;  in  other  words,  A  LIGHT  RAY 

TRANSMITTED  ALONG  AN  OPTIC  AXIS  HAS  THE  INTERMEDIATE 
REFRACTIVE  INDEX  /?. 

To  the  points  C,C  of  the  circular  section,  which  lie  in  the 
principal  section,  there  corresponds  the  ray  Oc,  which  stands 
perpendicular  to  the  normals  erected  on  the  index-surface  at  C 
and  C;  of  this  ray,  therefore,  the  vibration  direction  within  the 
crystal  lies  in  the  principal  section  in  question,  and  after  emerg- 
ing through  that  boundary-plane  of  the  crystal  that  is  parallel  to 
the  circular  section  (whereat  the  bi-normal  cone  is  transformed 
into  a  cylinder)  the  ray  vibrates  parallel  to  Me,  perpendicular 
therefore  to  the  vibration  direction  of  the  ray  OM.  To  any  and 
every  point  lying  between  Y  and  C  on  the  circular  section,  e.g. 
R,  there  corresponds  a  ray  Or  of  the  bi-normal  cone;  this  ray  is 


156  OPTICALLY   BIAXIAL   CRYSTALS 

perpendicular  to  the  normal  erected  on  the  index-surface  at  R, 
and  therefore  its  vibrations  take  place  in  the  plane  ROM  passing 
through  the  front- normal.  Since  the  face  of  emergence  of  the 
light,  which  face  is  parallel  to  the  ray-front,  intersects  the  bi- 
normal  cone  in  a  circle  passing  through  Mrc,  and  since,  from 
here  on,  all  rays  of  this  cone  are  transmitted  in  the  air  and  paral- 
lel to  OM,  as  a  ray  cylinder,  the  vibration  direction  of  the  ray 
Or  in  the  air  is  parallel  to  Mr  (its  polarization  plane  being 
Ore).  Accordingly,  for  any  ray  Or  of  the  cylinder  we  obtain 
the  vibration  direction  if  we  connect  r  and  M  by  a  straight  line; 
consequently  the  rays  on  the  periphery  of  the 
cylinder  have  indeed,  as  was  already  mentioned 
above,  all  possible  vibration  azimuths  between 
Me  —  that  of  the  ray  Oc  —  and  the  azimuth  per- 
pendicular to  Me  —  that  of  the  ray  OM.  (These 
g'  78'  vibration  directions  for  the  rays  corresponding 
to  different  points  on  the  circle  Mer  are  sketched  in  Fig.  78, 
with  a  < — >.) 

The  last-mentioned  fact  has,  as  its  consequence,  that  be- 
tween crossed  nicols  a  biaxial  crystal  plate  cut  perpendicular  to 
one  of  the  two  optic  axes  behaves  in  an  essentially  different 
manner  from  a  plate  cut  from  a  uniaxial  crystal  perpendicular 
to  its  axis.  For  a  thin  bundle  of  parallel  rays  falling  on  the 
former  plate  is  transformed  within  the  crystal  into  a  cone  of 
diverging  rays,  and  this  cone,  on  emerging,  into  a  circular  cylin- 
der; so  that,  instead  of  a  small  bright  spot,  a  ring  of  light*  must 
appear,  and  on  the  circumference  of  this  ring  there  must  exist 
all  possible  vibration  azimuths.  With  the  analyzing  nicol  one 
can  extinguish  only  those  rays  of  the  ring  of  light  that  vibrate 
perpendicular  to  the  principal  section  of  the  nicol,  of  all  the 
rest  only  the  corresponding  component.  Between  crossed  nicols 

*  As  to  how  this  ring  of  light  arising  from  the  image  of  a  small  aperture 
may  be  made  visible  under  the  microscope,  see  Kalkowsky,  Zeitschr.  f.  Kryst.  u. 
Min.  1884,  9,  486;  further,  Liebisch,  Nachr.  Ges.  d.  Wissensch.  Gottingen,  1888, 
Nr.  5.  The  contrivance  suggested  by  the  latter  is  supplied  by  the  optical  works 
of  R.  Fuess,  in  Steglitz  bei  Berlin. 


INTERFERENCE   PHENOMENA   IN   PARALLEL   LIGHT       157 

every  infinitely  thin  ray  bundle  that  has  passed  through  a  biaxial 
crystal  parallel  to  an  optic  axis  yields  a  ring  of  light,  and  this 
ring  (when  the  vibration  directions  of  the  nicols  are  parallel  and 
perpendicular  to  Me,  Fig.  77)  is  broken  at  the  points  M  and  c, 
because  there  complete  extinction  takes  place.  When  such  a  crys- 
tal plate  perpendicular  to  an  optic  axis  is  illuminated  throughout 
its  whole  extent  by  rays  falling  normal  to  it,  all  these  incomplete 
light  rings  cover  one  another,  and  the  whole  plate  appears  bright. 
While,  therefore,  on  rotation  in  its  own  plane  between  crossed 
nicols  a  uniaxial  crystal  plate  traversed  by  light  rays  parallel 
to  the  optic  axis  is  in  all  positions  dark,  a  biaxial  crystal 
plate  under  the  same  circumstances  remains,  on  the  contrary, 
bright. 

On  the  other  hand,  every  plate  of  any  other  orientation  has 
two  mutually  perpendicular  vibration  directions  —  to  be  found 
by  the  construction  given  on  page  154  — and  therefore  be- 
tween crossed  nicols  appears  dark  so  often  as  these  directions 
are  parallel  to  the  principal  sections  of  the  two  nicols.  In  the 
special  case  where  the  plane  of  the  biaxial  crystal  plate  coincides 
with  one  of  the  three  principal  optic  sections,  the  above-men- 
tioned construction  shows  in  a  very  simple  manner  that,  as  also 
follows  directly  from  a  consideration  of  the  index-surface,  the 
vibration  directions  of  the  plate  are  the  two  axes  of  the  index- 
surface  lying  parallel  to  the  plate;  in  other  words,  the  two  prin- 
cipal vibration  directions  lying  in  the  plane  of  the  plate.  When 
the  plate  is  cut  parallel  to  only  one  axis  of  the  index-surface,  but 
inclined  to  the  others,  the  direction  of  rays  falling  normal  to  it  lies 
in  a  principal  optic  section;  and  here  also,  from  considerations 
(see  p.  130  et  seq.)  even  previous  to  the  construction  described 
on  page  154,  it  is  manifest  that  one  of  the  vibration  directions 
of  the  plate  is  the  axis  of  the  index-surface  (hence  a  principal 
vibration  direction)  to  which  the  plate  is  parallel,  the  other  the 
straight  line  drawn  in  the  plane  of  the  plate  normal  to  that  axis. 

Consequently,  if  into  the  orthoscope  we  bring  any  plate  of  a 
biaxial  crystal  that  is  not  cut  perpendicular  to  one  of  the  optic 


158  OPTICALLY   BIAXIAL   CRYSTALS 

axes,  and  rotate  it  between  crossed  nicols  in  its  own  plane, 
it  becomes  dark  four  times  and  in  the  intermediate  positions 
bright,  thus  exhibiting  the  interference  phenomena  of  mono- 
chromatic light,  considered  on  page  60.  The  phenomena  to 
be  observed  in  white  light  depend,  in  the  first  place,  on  whether 
the  vibration  directions  of  the  rays  of  different  color  emerging 
from  the  plate  are  the  same  or  different.  When  the  crystallo- 
graphic  orientation  of  the  axes  of  the  index-surface  is  for  all 
colors  the  same,  it  follows  directly  that  the  vibration  directions 
%also,  of  a  plate  parallel  to  one  or  to  two  of  the  axes  of  the  index- 
surface,  coincide  for  all  colors.  A  plate  oriented  parallel  to 
none  of  the  three  axes  of  the  index-surface  exhibits,  on  the  other 
hand,  a  dispersion  of  the  vibration  directions;  for  in  this  case  the 
construction  mentioned  on  page  154  yields,  because  of  the  in- 
equality of  the  optic  axial  angles  for  different  colors,  for  light 
of  another  wave  length  diverging  vibration  directions  as  well. 
Since,  however,  the  difference  among  the  axial  angles  seldom 
exceeds  a  few  degrees,  the  angles  between  the  vibration  direc- 
tions of  such  a  crystal  plate,  for  differently  colored  light,  are 
usually  very  small. 

But,  when  the  crystallographic  orientation  of  the  principal 
vibration  directions  for  different  colors  is  itself  different,  then 
the  vibration  directions  diverge,  even  of  such  plates  as  are  paral- 
lel to  one  or  to  two  of  the  axes  of  the  index-surface  for  a  certain 
color.  Accordingly,  if  we  desire  here  to  determine  the  vibra- 
tion directions  by  means  of  the  positions  of  darkness  in  the 
orthoscope,  in  the  way  mentioned  on  page  76,  this  must  be 
done  in  monochromatic  light,  and  specially  for  each  color.  A 
more  exact  method  of  determination  is  that  of  the  so-called 
stauroscope. 

This  method  depends  on  the  principle  that,  when  the  crystal  to  be  investi- 
gated has  the  orientation  corresponding  to  the  maximum  extinction  of  the 
light,  it  effects  no  resolution  of  the  light  passing  through  it  and  therefore  does 
not  disturb  the  formation  of  an  interference-figure  produced  by  another 
crystal,  inserted  in  the  path  of  the  light  rays  and  illuminated  by  convergent 


INTERFERENCE   PHENOMENA   IN   PARALLEL   LIGHT       159 

light.  If  the  latter  crystal  is  suitably  chosen,  then,  when  the  vibration  direc- 
tions of  the  crystal  to  be  investigated  diverge  from  those  of  the  nicols 
even  very  slightly,  there  results  a  perceptible  deformation  of  the  observed 
interference-figure;  and  hence  the  adjustment  can  be  made  exceedingly 
sensitive-* 

A  biaxial  crystal  plate  from  which  the  light  rays  of  all  colors 
emerge  with  the  same,  or  at  least  very  nearly  the  same,  vibra- 
tion directions  will  obviously,  even  with  the  use  of  white  light, 
during  one  complete  rotation  in  the  orthoscope  appear  dark  four 
times,  but  in  the  intermediate  positions  colored,  if  it  is  thin; 
for  the  considerations  on  page  107  as  to  the  phenomena  with  a 
uniaxial  crystal  plate  cut  oblique  to  the  optic  axis  manifestly 
apply  to  every  doubly  refracting  body.  The  arising  color  must 
vary  with  the  thickness  of  the  plate,  and  with  a  certain  thick- 
ness of  the  same  pass  over  into  the  white  of  a  higher  order. 
Plates  of  a  different  orientation  exhibit,  in  general,  different 
interference-colors  even  with  equal  thickness;  and  hence,  with 
them  also,  the  phenomenon  of  the  white  of  a  higher  order 
appears  with  different  plate  thickness.  It  appears  with  the 
least  thickness  in  the  case  of  a  plate  parallel  to  the  optic  axial 
plane;  for  in  such  a  plate  the  light  is  split  up  into  two  rays 
of  which  one  has  the  greatest,  the  other  the  least  transmission 
velocity,  the  double  refraction  (corresponding  to  the  difference 
between  the  principal  refractive  indices  f  and  a)  thus  being 
altogether  the  strongest  in  the  crystal.  The  double  refraction  is 
weaker  in  plates  parallel  to  the  two  other  principal  sections,  and 
its  strength  is  different  for  the  two  orientations,  according  to  the 
magnitude  of  the  differences  7-  —  /?  and  /?  —  a.  While  with- 
uniaxial  crystals  all  plates  forming  equal  angles  with  the  optic 
axis  have  the  same  birefringence,  and  consequently  with  equal 
thickness  exhibit  the  same  color,  there  exists  no  such  uniformity 
around  an  optic  axis  of  a  biaxial  crystal;  for  if,  starting  out  from 
a  circular  section  of  the  index-surface  as  the  plane  of  the  plate, 

*  For  the  description  of  such  instruments  and  the  procedure  in  stauroscopic 
measurements,  cf.  footnote  f,  p.  30. 


l6o  OPTICALLY   BIAXIAL   CRYSTALS 

we  approach  the  one  or  the  other  of  the  principal  sections,  we 
always  find  different  ellipses  as  intersections  of  the  index-surface, 
and  hence,  also,  different  strength  of  the  double  refraction. 

In  case  the  vibration  directions  of  a  biaxial  crystal  plate  for 
different  colors  diverge  considerably,  then  in  white  light  the 
plate  appears  in  no  position  absolutely  dark;  for  when  it  has 
been  rotated  in  its  own  plane  to  the  position  of  extinction  for  a 
definite  color,  the  remaining  colors  still  pass  through  — with  an 
intensity  that  is  the  greater,  the  greater  the  dispersion  of  the 
vibration  directions  —  and  therefore  make  the  plate  appear 
bright,  bright  with  a  composite  color.  With  exceptional  size  of 
the  angle  between  the  vibration  directions  of  different  colors 
there  appears,  on  rotating  the  plate  in  the  orthoscope,  a  contin- 
uous change  of  very  vivid  colors. 

A  certain  similarity  to  the  phenomena  resulting  from  the 
dispersion  of  the  vibration  directions  is  possessed  by  those  ex- 
hibited in  white  light  by  a  biaxial  crystal  plate  cut  perpendicular 
to  an  optic  axis.  Here,  for  that  color  for  which  the  plate  stands 
exactly  perpendicular  to  the  optic  axis,  there  exist,  in  conse- 
quence of  the  interior  conical  refraction  (as  we  saw  on  p.  156), 
all  possible  vibration  azimuths;  on  rotation,  therefore,  bright- 
ness is  observed  in  every  position.  For  all  the  remaining  colors 
the  optic  axis  stands  more  or  less  oblique  to  the  plate,  so  that  for 
them  a  resolution  takes  place  into  two  mutually  perpendicular 
vibrations;  but,  for  the  different  colors,  these  vibrations  are 
differently  oriented.  Therefore  in  white  light,  also,  the  plate 
can  in  no  position  exhibit  darkness. 

In  the  microscopical  investigation  of  rocks  this  last-men- 
tioned behavior  of  biaxial  crystals  often  permits  a  mineral  ap- 
pearing in  a  thin  rock  section  in  numerous  diversely  oriented 
sections  of  the  mineral  to  be  recognized  as  biaxial.  When  these 
mineral  sections  exhibit  very  different  birefringence  (those  with 
the  highest,  according  to  p.  159,  are  approximately  parallel  to  the 
optic  axial  plane) ,  then,  if  among  them  there  chance  to  be  some 
that  on  the  rotation  between  crossed  nicols  always  remain 


INTERFERENCE   PHENOMENA   IN   CONVERGENT   LIGHT     161 

bright  and  at  the  most  exhibit  changing  colors,  these  sections,  in 
the  cutting,  must  have  been  struck  perpendicular  to  an  optic 
axis.  Absolute  certainty  that  one  has  to  do  with  this  case  and 
not  with  that,  previously  considered,  of  a  very  great  dispersion  of 
the  vibration  directions,  is  obtained  if  one  brings  such  a  section 
into  the  center  of  the  field  and  makes  in  the  microscope  the  \ 
changes  through  which  convergent  light  is  produced.  Through 
the  crystal  section  one  then  beholds  those  phenomena  of  a  plate 
cut  perpendicular  to  an  optic  axis  that  are  to  be  discussed  in 
the  next  section. 

INTERFERENCE  PHENOMENA  OF  BIAXIAL  CRYSTALS  IN 
CONVERGENT  POLARIZED  LIGHT 

Like  those  cut  oblique  to  the  axis  from  uniaxial  crystals,  thin 
plates  of  biaxial  crystals  exhibit  in  monochromatic  light  dark 
and  bright  curves;  the  points  of  such  a  curve  correspond  to 
the  rays  that  in  the  crystal  have  acquired  the  same  difference 
of  path.  When  the  plate  is  cut  parallel  to  the  optic  axial  plane, 
these  curves  have  the  form  of  hyperbolas,  as  with  a  uniaxial 
plate  parallel  to  the  optic  axis.  (See  p.  120.)  In  white  light  there 
appear  analogous  curves,  each  one  of  a  uniform  color  (isochro- 
matic  curves) ;  with  excess  of  a  certain  thickness,  however,  only 
the  white  of  a  higher  order  is  visible,  unless  there  fall  within  the 
field  of  the  conoscope  rays  that  have  passed  through  the  crystal 
parallel  to  an  optic  axis. 

To  elucidate  the  phenomena  arising  in  the  case  last  men- 
tioned, let  us  consider  first  the  behavior  in  monochromatic  light 
of  a  crystal  plate  cut  perpendicular  to  an  optic  axis  (for  the 
color  in  question);  in  such  a  plate  the  rays  that  have  passed 
through  parallel  to  this  axis  converge  at  the  center  of  the  field. 
If  one  turns  the  plate  in  its  own  plane  so  that  its  optic  axial  plane 
is  parallel  to  the  vibration  direction  of  one  of  the  two  crossed 
nicols,  then,  in  the  field  of  view,  parallel  to  this  principal  sec- 
tion there  must  appear  a  dark  bar,  since  in  the  principal  section 
there  now  take  place  only  such  vibrations  as  in  the  second  nicol 


162  OPTICALLY  BIAXIAL   CRYSTALS 

are  totally  annihilated.  Points  on  the  two  sides  of  the  dark  bar 
correspond  to  rays  whose  vibration  directions  are  more  and 
more  inclined  to  the  principal  section;  consequently,  as  we  pass 
out  from  the  middle  line  of  the  bar  there  gradually  takes  place  a 
brightening.  And  this  brightening  goes  on  the  most  rapidly 
near  the  center  of  the  field;  because  here,  since  this  point 
corresponds  to  the  optic  axis,  the  vibration  direction  varies  the 
most  rapidly,  and  because  in  immediate  proximity  to  the  axial 
point  there  come  in  addition  the  rays  of  the  conical  refraction, 
with  their  vibrations  taking  place  in  all  azimuths.  So  the  bar 
traversing  the  field  of  view  is  narrowest  at  the  center.  At  a  cer- 
tain distance  from  the  center  of  the  field  the  rays  will  converge 
that  in  the  crystal  have  acquired  a  path  difference  amounting  to 
one  wave  length  of  the  color  in  question;  at  this  distance  there 
must  occur  annihilation  of  the  light,  and  therefore  the  center 
must  be  surrounded  by  a  closed  curve  of  darkness.  Unlike  the 
dark  rings  of  the  uniaxial  crystals  (see  p.  112  et  seq.),  however, 
this  curve  cannot  have  the  form  of  a  circle.  For  in  the  direc- 
tion parallel  to  the  dark  bar  the  path  difference  increases  from 
the  center,  where  it  is  zero,  on  the  two  sides  of  the  center  un- 
equally (in  the  one  case  the  birefringence  approaching  the 
value  f  —  /?,  in  the  other  /?  —  «),  and  on  the  two  sides  of  the 
black  bar  the  arrangement,  although  symmetrical  with  reference 
to  the  bar,  is  different  in  every  azimuth.  The  dark  ring  there- 
fore resembles  an  ellipse  in  form  and  is  symmetrically  bisected 
by  the  dark  bar.  The  same  applies  to  the  second  ring  sur- 
rounding the  center,  which  corresponds  to  the  path  difference 
2L  And  so  on.  When  the  plate  is  cut  oblique  to  an  optic 
axis,  the  center  of  the  bright  and  the  dark  rings  does  not  appear 
at  the  center  of  the  field,  and  on  account  of  the  refraction  the 
rings  are  somewhat  different  in  form. 

Under  some  circumstances  the  ring  systems  even  of  both 
optic  axes  may  fall  in  the  field  of  the  conoscope;  — and  of 
practical  importance,  here,  are  especially  those  interference  phe- 
nomena that  are  exhibited  by  a  plate  whose  plane  stands  per- 


INTERFERENCE    PHENOMENA   IN    CONVERGENT   LIGHT      163 


pendicular  to  the  acute  bisectrix  (the  bisector  of  the  acute  angle 
of  the  optic  axes  — see  p.  138).  If  with  crossed  nicols  we  view 
such  a  plate  in  monochromatic  light  and  in  a  position  where  its 
optic  axial  plane  is  parallel  to  the 
polarization  plane  of  one  of  the 
two  nicols,  we  behold  the  fol- 
lowing phenomenon  (Fig.  79) : 
Through  the  center  of  the  field 
there  passes  a  black  cross,  whose 
two  opposite  arms  lying  parallel 
to  the  axial  plane  appear  far  nar- 
rower and  more  sharply  defined 
than  do  the  two,  of  less  regular 
form,  that  stand  perpendicular  to  them.  The  two  points  — 
they  lie  equidistant  from  the  center  of  the  field  on  the  two  sides 
—  at  each  of  which  the  rays  passing  through  the  crystal  along 
an  optic  axis  converge,  are  surrounded  by  dark  and  by  bright 
oval  rings;  and  two  of  these  rings,  of  a  certain  distance  from  the 


Fig.  79- 


Fig.  80. 


Fig.  81. 


two  centers  respectively,  have  together  the  form  oo ,  while 
those  of  greater  distance  are  shaped  like  the  outermost  curves 
represented  in  Fig.  79.  These  curved  lines,  in  the  case  before 
us  lines  of  equal  brightness,  are  called  lemniscates.  If  with- 
out altering  the  crossed  position  of  the  nicols  one  rotates  the 
crystal  plate  in  its  own  plane,  the  rings  do  not  change  in  the 


164  OPTICALLY  BIAXIAL  CRYSTALS 

least,  but  simply  rotate  with  the  plate;  the  arms  of  the  cross, 
on  the  other  hand,  previously  straight,  become  transformed  into 
two  hyperbolas,  which  with  slight  rotation  of  the  plate  appear  as 
in  Fig.  80  (p.  163)  and  with  45°  rotation  as  in  Fig.  81,  but  which 
therewith  always  pass  through  the  two  centers  of  the  ring 
systems.  These  two  centers,  the  so-called  "axial  points",  are 
the  nearer  to  the  center  of  the  field,  the  smaller  the  optic  axial 
angle;  the  closer  to  the  edge,  the  larger  that  angle:  their  dis- 
tance from  each  other  is  a  measure  of  that  angle.  Since  the 
optic  axial  angle  (for  a  definite  color)  is  with  all  crystals  of  a 
substance  the  same,  so  also  does  the  distance  between  the  centers 
of  the  two  ring  systems  remain  the  same,  whether  the  plate 
be  thick  or  thin,  provided  only  that  it  consists  of  the  same 
material. 

The  above-described  phenomena  are  explained  as  follows :  — • 
In  Fig.  82  *  are  sketched  with  crosses,  for  numerous  points 
in  the  field  of  view,  the  two  vibration  directions  that   are  ob- 
tained,   by    the    construction    described 
on  page  1^4,  for  the  transmission  direc- 

r     °  J 

tions  corresponding  to  these  points. 
Between  crossed  nicols  all  points  must 
appear  dark  at  which  the  vibration  direc- 
tions  coincide  with  those  of  the  nicols. 
When  the  latter  are  horizontal  and  ver- 

xy.-/.-yLf -f-t--t--Hr-vf  .«       i  i  i        i  i 

^tiittU**'  *lca*  a  black  cross  must  appear,    whose 

Fig.  82.  vertical  arms  are   very  broad,  especially 

toward  the  edge  of  the  field,  because  on 

the  two  sides  of  the  vertical  middle  line  the  obliqueness  of  the 
vibration  directions  grows  only  very  gradually;  the  same  is  the 
case  also  for  the  extreme  parts  of  the  horizontal  arms,  while 
above  and  below  the  axial  points  the  obliqueness  of  the  vibra- 
tion directions  increases  very  rapidly.  So  the  horizontal  dark 
bar  must  be  narrowest  near  the  axial  points,  and  in  both  direc- 
tions from  the  same  increase  in  breadth.  Hence,  if  we  imagine 
*  Copy  after  E.  G.  A.  ten  Siethoff,  Centralbl.  f.  Min.  1890,  268. 


INTERFERENCE    PHENOMENA   IN   CONVERGENT   LIGHT     165 

Fig.  82  as  rotated*  a  small  angle  to  the  right  (i.e.  clockwise), 
and  connect  the  points  at  which  the  vibration  directions  now 
stand  horizontal  and  vertical,  thus  coinciding  with  those  of  the 
nicols,  we  obtain  the  dark  curves  shown  in  Fig.  80;  after  a 
rotation  of  45°,  the  two  hyperbolas  as  in  Fig.  81;  and  these 
curves  must  necessarily  have  their  least  breadth  at  the  axial 
points. 

If  from  these  axial  points,  at  which  there  is  no  double  re- 
fraction and  consequently  no  difference  of  path,  we  pass  out  in 
a  direction  that  is  not  parallel  to  the  optic  axial  plane,  then  at  a 
certain  distance  those  rays  will  converge  that  interfere  with  ?A 
difference  of  path;  at  a  greater  distance,  those  interfering  with 
^  difference  of  path;  and  so  on.  Advancing  in  this  direction, 
one  must,  in  the  interference-figure,  come  alternately  upon 
minima  and  maxima  of  brightness.  But  if  we  vary  the  direc- 
tion in  which  we  proceed  from  the  center,  then  at  the  same  dis- 
tance there  is  a  variation  not  only  in  the  difference  in  velocity  of 
the  two  arising  rays,  but  also  in  the  distance  they  travel  in 
the  crystal;  we  thus  obtain  the  same  difference  of  path,  i.e.  the 
same  minimum  and  maximum,  at  a  different  distance  from  the 
center.  Therefore,  while  with  a  uniaxial  crystal  the  points  of 
equal  brightness  lie  on  circles,  — because  the  variation  of  the 
double  refraction  with  the  inclination  takes  place  in  the  same 
way  in  all  directions  about  the  axis, —  here  oval  curves  of  equal 
brightness  must  arise;  but,  since  the  distance  between  the 
two  skins  of  the  ray-surface  — this  distance  determining  the 
birefringence  — varies  symmetrically  on  the  two  sides  of  each 
principal  section  of  the  crystal,  these  ovals,  too,  must  be  sym- 
metrically bisected  by  the  optic  axial  plane  and  by  the  principal 
section  standing  perpendicular  to  that  plane;  i.e.  by  the  two  black 
bars.  As  a  matter  of  fact  it  follows  from  the  theory,  agreeably  to 
observation,  that  the  dark  and  the  bright  curves  have  the  form  of 

*  Instead  of  by  rotating  the  crystal  plate,  one  can  of  course  produce  the  same 
effect  by  rotating  the  two  crossed  nicols  together  while  the  plate  retains  its  original 
orientation. 


l66  OPTICALLY    BIAXIAL   CRYSTALS 

so-called  lemniscates,  which  fulfill  the  above-stated  conditions. 
If,  then,  we  rotate  the  plate  in  its  own  plane,  the  lemniscate  sys- 
tems likewise  must  rotate,  since  their  formation  is,  as  we  know, 
closely  associated  with  definite  directions  in  the  crystal  and 
since  the  line  connecting  their  centers  must  always  remain 
parallel  to  the  optic  axial  plane.  But  the  points  of  the  inter- 
ference-figure at  which  the  rays  converge  whose  vibration  direc- 
tions are  parallel  to  the  polarization  plane  of  a  nicol  —  these 
rays  therefore  being  totally  annihilated  — now  lie  no  longer  on 
two  straight  lines  crossing  each  other  at  right  angles,  but  on 
two  hyperbolas,  and  naturally  these  hyperbolas  must  each  pass 
through  an  axial  point. 

If  with  unchanged  color  of  the  light  used  we  employ  a 
thicker  plate  of  the  same  substance,  the  black  cross  or  the  hyper- 
bolas, as  also  the  distance  between  the  two  axial  points,  must, 
according  to  what  has  been  said  up  to  the  present,  remain 
wholly  unaltered.  And  to  be  sure,  at  any  given  distance  from 
an  axial  point  the  difference  in  velocity  of  the  two  doubly  re- 
fracted rays  will  indeed  be  the  same  as  before.  But,  since  at 
that  distance  the  path  of  these  two  rays  in  the  crystal  is  longer, 
their  "  path  difference  "  at  that  distance  must  be  greater;  and 
therefore,  in  the  position  where  in  the  case  of  the  thinner  plate 


Fig.  83. 


the  first  dark  ring  was  seen,  there  appears  with  the  thicker  plate 
even  the  second  ring  or  the  third.  Thus,  the  thicker  the  plate, 
the  less  will  be  the  width  of  the  rings;  the  thinner  the  plate,  the 
wider  the  rings.  Accordingly,  with  a  certain  slight  thickness  of 


INTERFERENCE  PHENOMENA  IN  CONVERGENT  LIGHT     167 

a  plate  having  a  small  optic  axial  angle,  the  case  will  arise  that 
even  the  innermost  lemniscate  no  longer  consists  of  two  sepa- 
rate ovals,  but  of  an  ellipse-like  form  surrounding  both  centers, 
like  one  of  the  outermost  lemniscates  appearing  with  a  thicker 
plate.  Figure  836  represents  the  interference-figure  of  a  plate 


Fig.  84. 

having  such  a  thinness,  compared  with  that,  Fig.  83  a,  of  a 
thicker  plate  of  the  same  substance,  both  under  parallelism  of 
the  axial  plane  with  the  principal  section  of  a  nicol;  while  in 
Fig's  84  a  and  b  is  given  the  interference-figure  of  the  same 
plates  after  a  rotation  of  45°. 

Since  the  width  of  the  rings  depends  on  the  difference  in  the 
velocity  with  which  the  two  rays  arising  by  the  double  refrac- 
tion are  transmitted  in  the  crystal,  their  width,  just  as  in  the 
case  of  optically  uniaxial  crystals,  is  different  with  different  sub- 
stances, even  when  the  plate  thickness  remains  the  same.  In 
other  words,  the  width  of  the  rings  depends  on  the  birefringence 
(the  difference  among  the  three  principal  refractive  indices  a, 
/?,  7-) :  with  a  plate  that  consists  of  a  substance  having  slight 
birefringence  the  rings  are  wider  than  are  exhibited  by  an 
equally  thick  plate  of  a  body  whose  double  refraction  is 
stronger. 

Finally,  as  follows  from  their  origin,  the  width  of  the  rings 
depends  further  on  the  wave  length  of  the  light  used:  when 
this  is  greater  the  rings  stand  farther  apart  from  one  another, 
and  vice  versa.  So  the  dark  rings  for  the  different  colors  fall 
in  different  positions;  and  consequently,  if  the  plate  is  investi- 


1 68  OPTICALLY   BIAXIAL   CRYSTALS 

gated  in  while  light,  colored  rings  will  arise,  the  explanation  of 
which  is  wholly  the  same  as  with  the  uniaxial  crystals.  But, 
while  with  the  latter  crystals  the  dark  rings  for  the  different 
colors  simply  overlap  one  another  as  concentric  circles,  the 
isochromatic  curves  thus  being  again  circles  having  the  same 
center  (the  locus  of  the  optic  axes),  with  biaxial  crystals  the 
axial  points  for  the  different  colors  do  not  coincide,  because  the 
optic  axial  angles  to  which  they  correspond  are  not  equal.  As  a 
result,  the  color  phenomena  are  more  complex;  but  this  in  a 
different  degree  according  as  the  cr'ystallographic  directions  of 
the  greatest,  the  least,  and  the  intermediate  light  velocity  for 
the  different  colors  coincide  or  not.  With  those  biaxial  crystals 
in  which  these  three  directions  are  different  for  different  colors, 
the  same  is  accordingly  true  also  of  the  axial  planes.  There- 
fore a  plate  cut  from  such  a  crystal  can  stand  perpendicular  to 
the  acute  bisectrix  only  for  one  color,  and  consequently  when 
only  this  is  used  the  lemniscate  system  will  have  for  its  center 
the  exact  center  of  the  field;  but  if  we  illuminate  with  a  different 
color,  the  rings  will  not  only  have  a  different  width  and  their 
centers  be  a  different  distance  apart,  but  the  whole  figure  will  be 
shifted  in  the  field.  In  white  light,  therefore,  color  curves  will 
arise  that  are,  to  be  sure,  similar  to  the  lemniscates  when  the 
divergence  of  the  bisectrices  for  different  colors  amounts  at  the 
most  to  a  few  degrees,  — as  is  usually  the  case,  —  but  in  which 
the  color  sequence  is  in  detail  unsymmetrical,  so  that  the  right 
side  is  not  symmetrical  with  the  left,  nor  the  upper  side  with 
the  lower. 

Turning  our  attention  first  of  all  to  the  simplest  case,  in 
which  the  three  principal  vibration  directions,  and  consequently 
also  the  optic  axial  plane,  have  for  all  colors  the  same  crys- 
tallographic  orientation,  the  interference-figure  for  each  single 
color  must  be  symmetrically  bisected  both  by  the  straight  line 
connecting  the  two  axial  points  and  by  the  straight  line  inter- 
secting this  first  line  perpendicularly  at  the  center  of  the  field; 
i.e.  by  the  two  black  bars  that  appear  when  the  axial  plane  is 


INTERFERENCE   PHENOMENA   IN   CONVERGENT   LIGHT     169 


parallel  to  the  principal  section  of  a  nicol.  Since  these  lines  of 
symmetry  coincide  for  all  colors,  there  appears  in  white  light 
an  interference-figure  that  likewise  is  symmetrically  bisected  by 
those  same  straight  lines;  in  other  words,  whose  upper  half  is 
exactly  equal  to  the  lower  (in  the  reverse  position)  and  whose 
right  side  is  in  like  manner  equal  to  the  left.  (See  Fig.  3  of 
Plate  II.)  If  in  this  figure  we  compare  the  different  parts  of  a 
lemniscate,  e.g.  of  the  first  color  ring  surrounding  one  of  the 
optic  axes,  we  see  that  they  are  not  colored  alike:  that  the  side 
(also  of  the  succeeding  rings)  turned  toward  the  center  of  the 
field  has  a  different  color  sequence  from  that  turned  outward, 
while  the  upper  half  is 
exactly  equal  and  op- 
posite to  the  lower. 
This  is  explained 
simply  by  the  circum- 
stance that  the  centers 
of  the  ring  systems  for 
the  different  colors  do 
not  coincide.  Let,  for 
example,  A  A  (Fig.  85) 


Fig.  85. 


be  the  direction  of  the 
axial  plane  (for  all 
colors),  BB  that  of  the  principal  section  standing  perpen- 
dicular to  it;  and  let  r,r  represent  the  two  axial  points  for  red, 
b,b  those  for  blue,  the  distance  rb  being  of  course  the  same  on 
both  sides,  because  the  bisectrices  for  the  two  colors  coincide 
at  the  center  of  the  figure.  Then  may  the  curves  drawn 
full  be  the  dark  lemniscates  for  red,  the  dotted  curves  those 
for  blue.  Hence,  as  the  figure  shows,  if  we  pass  out  from 
the  center  of  one  of  the  two  ring  systems,  when  we  move 
toward  the  center  of  the  figure  blue  first  is  annihilated,  red 
only  at  a  greater  distance;  but  when  we  advance  outward  the 
extinction  takes  place  in  the  reverse  order.  Thus,  in  white 
light  and  with  such  a  crystal,  the  color  sequence  of  the  inner- 


1 70  OPTICALLY   BIAXIAL    CRYSTALS 

most  ring  must  in  these  two  directions  be  exactly  the  opposite; 
if  the  distance  between  (dispersion  of)  the  optic  axes,  rb,  is 
not  so  great,  this  color  sequence  in  the  two  directions  must  at 
least  be  different.  Figure  85  shows  further  that  the  upper  half 
of  the  color  rings  must  be  exactly  equal  and  opposite  to  the 
lower,  because  the  shifting  takes  place  exactly  along  the  straight 
line  AA\  and  that  the  right-hand  ring  system  must  likewise 
be  equal  and  opposite  to  the  left,  because  the  opposite  shifting 
of  the  rings  on  the  two  sides  of  the  line  BB  must  always  proceed 
in  the  same  way,  so  that  this  line  exactly  bisects  the  systems  for 
all  colors.  The  phenomena  must  accordingly  appear  as  repre- 
sented by  Fig.  3,  Plate  II,  in  which  the  sides  turned  toward  each 
other,  of  the  two  innermost  color  rings,  are  colored  alike,  while 
those  turned  away  from  each  other  are  likewise  colored  alike 
but  in  a  different  way.  In  case  the  points  r  and  b  (Fig.  85)  are 
at  such  a  distance  from  each  other  that  the  first  dark  ring  for 
blue  falls  in  largest  part  or  entirely  outside  that  for  red,  no 
lemniscate-like  color  curves  are  formed  at  all,  but  an  interfer- 
ence-figure arises  which  approximates  to  that  of  brookite  and 
others.  To  these  further  reference  will  be  made  on  page  172. 

If  one  turns  the  crystal  plate  in  its  own  plane  so  that  the 
axial  plane  forms  an  angle  of  45°  with  the  nicols,  then,  as  we 
know,  there  appear  the  black  hyperbolic  brushes  passing  through 
the  axial  points.  As  in  Fig.  85,  so  also  in  Fig.  86,  which  corre- 
sponds to  this  position,  let  AA  be  the  axial  plane,  r,r  and  b,b  the 
axial  points  for  red  and  blue,  the  full  and  the  dotted  curves  the 
dark  lemniscates  for  these  colors;  then  it  is  seen  that  the  color 
rings  arising  in  white  light  must  be  wholly  the  same  as  arose  in 
the  former  position  of  the  plate.  But  if  we  consider  the  dark 
hyperbolas,  we  see  that  the  curves  for  the  different  colors  can- 
not coincide,  since  the  two  arising  for  each  color  must  pass 
through  the  axial  points.  In  the  figure  the  two  hyperbolas 
appearing  in  red  light  are  hatched  vertically,  those  for  blue 
horizontally;  and  therefrom  it  is  manifest  that  the  curves  for  the 
different  colors  (since  for  the  remaining  colors  they  lie  between 


INTERFERENCE   PHENOMENA   IN   CONVERGENT   LIGHT      171 


those  mentioned)  partially  overlap  one  another,  overlapping  the 
more,  the  less  the  distance  rb,  —  i.e.  the  less  the  optic  axial 
angles  for  the  different  colors  differ  from  one  another.  At  the 
points  where  the  component  hyperbolas  for  all  colors  fall  one 
upon  the  other,  i.e.  along  the  middle  of  the  hyperbolic  stripes, 
total  darkness  will  arise;  not  so,  however,  at  the  two  edges, 
where  the  extinction  takes  place  only  for  a  part  of  the  colors. 
Therefore  the  borders  must  be  edged  with  color,  as  may  be 
seen  from  Fig.  4  of  Plate  II,  while  the  black  bars  appearing 
with  the  first  position  of  the  crystal  plate  (Fig.  3,  ibid.)  can 
exhibit  nothing  of  the  kind.  The  color  edging  of  the  dark 
hyperbolas  is  the  broader  and  the  more  vividly  colored,  the 
fewer  the  number  of  the  component  hyperbolas  that  fall  one 
upon  the  other  at  any  one  point;  i.e.  the  greater  the  dispersion 
of  the  axes  —  the  variation  of  the  axial  angle  for  the  different 
colors.  When  the  dispersion  attains  such  a  magnitude  that  even 
along  the  middle  of  the  hyperbolic  stripes  the  curves  no  longer 
overlie  one  another  for  all  colors, 
no  black  appears  even  there:  the 
hyperbolas  consist  only  of  color 
stripes,  in  a  definite  sequence 
from  within  outward.  This 
sequence  must  at  the  same  time 
show  the  sense  of  the  dispersion 
of  the  axes;  that  is  to  say, 
whether  the  axial  angle  for  the 
rays  at  the  red  end  of  the 
spectrum  is  smaller  than  for 
those  of  the  violet  part  (briefly 
expressed:  p  <  u),  or  vice 
versa  (p>u).  In  Fig.  86,  and  in  Fig.  4  of  Plate  II,  is  repre- 
sented the  interference-figure  of  a  crystal  of  the  former  kind. 
In  this  figure,  on  the  two  hyperbolic  curves  passing  through  the 
points  r,r,  the  red  light  is  totally  extinguished,  and  so  too  are 
the  parts  of  the  spectrum  adjacent  to  the  red  (or  at  least 


Fig.  86. 


!72  OPTICALLY   BIAXIAL   CRYSTALS 

much  weakened  in  intensity);  but  not  the  colors  of  the  other 
end,  namely  blue  and  violet,  which  are  extinguished  only  when 
we  reach  the  points  designated  by  b,b.  Consequently  these 
two  colors  will  appear  as  an  edging  of  the  two  hyperbolas  on  the 
side  turned  toward  the  line  BB,  while  on  the  concave  side, 
turned  outward,  a  red  edging  will  appear,  because  here  the  blue 
rays  are  totally  annihilated.  These  color  edgings  always  stand 
out  the  most  distinctly  on  those  parts  of  the  hyperbolas  that 
lie  within  the  innermost  color  ring,  where  the  hyperbolas  are 
most  sharply  bounded.  If  one  there  observes  that  the  inner 
side  (i.e.  the  side  turned  toward  the  center  of  the  field)  of  the 
hyperbolas  is  colored  blue,  the  outer  side  red  (see  Fig.  4,  Plate 
II),  one  has  to  do  with  a  crystal  whose  optic  axial  angle  for  red 
is  smaller  than  for  blue  (sense  of  the  dispersion,  p  <  u) ;  when 
on  the  other  hand  the  inner  side  is  red  and  the  outer  blue,  the 
sense  of  the  dispersion  is  p  >  u  —  the  axial  angle  for  red  larger 
than  for  blue.  The  broader  and  the  more  vivid  the  color 
edgings,  the  greater  is  the  strength  of  the  dispersion.  Since 
with  all  crystals  of  one  and  the  same  substance  the  optic  axial 
angle  for  the  same  color  is  identical,  this  holds  true  also  both  of 
the  sense  and  of  the  strength  of  the  dispersion. 

It  is  not  difficult  to  understand  that  a  plate  whose  faces  stand 
perpendicular  to  the  bisector  of  the  obtuse  axial  angle  must 
exhibit  wholly  analogous  interference  phenomena,  i.e.  lemnis- 
cate  systems,  but  that,  of  these  systems,  the  centers  correspond- 
ing to  the  two  axes  will  usually  stand  so  far  apart  from  each 
other  that  they  no  longer  fall  in  the  field  of  the  instrument. 
When  they  do,  however,  the  color  edgings  of  the  dark  hyper- 
bolas must  of  course  indicate  the  opposite  sense  of  dispersion  to 
that  with  the  plate  perpendicular  to  the  acute  bisectrix. 

In  some  substances,  as  brookite  (TiO2),  ammonium  mellitate, 
and  others,  the  three  principal  refractive  indices  vary  with  the  color 
so  diversely  that,  for  example,  the  index  that  for  the  colors  lying  at 
one  end  of  the  spectrum  is  the  smallest  becomes,  for  a  certain 
color,  equal  to  the  intermediate,  and  for  a  vibration  period  that  is 


INTERFERENCE   PHENOMENA   IN   CONVERGENT   LIGHT     173 

still  more  different  these  two  indices  exchange  rdles.  In  crystals 
having  such  strong  dispersion  the  axial  plane  for  one  part  of  the 
spectrum  must  be  one  principal  section,  and  that  for  another 
part  one  of  the  other  two,  the  latter  axial  plane  thus  standing 
perpendicular  to  the  former;  and  for  a  certain  color  lying  inter- 
mediate in  the  spectrum  the  crystal  must  be  uniaxial*  When 
the  acute  bisectrix  coincides  for  all  colors,  then,  if  one  brings 
into  convergent  polarized  light  a  plate  cut  perpendicular  to  this 
bisectrix,  one  sees  in  red  light  an  ordinary  interference-figure, 
and  in  blue  light  likewise  such  a  figure  but  with  the  axial  plane 
oriented  perpendicular  to  that  of  the  former  figure;  in  white 
light,  on  the  other  hand,  an  absolutely  different  color  image 
appears,  which  is  represented  in  Fig.  5  on  Plate  II. 

In  the  crystals  hitherto  considered  the  three  principal  sec- 
tions of  the  index-surface,  or  of  the  ray-surface,  for  all  the  dif- 
ferent colors  coincide;  the  optical  properties,  therefore,  although 
they  do  vary  for  every  color  in  a  different  way,  always  vary 
symmetrically  with  respect  to  these  three  planes,  which  we  will 
accordingly  designate  as  planes  of  optical  symmetry  of  the 
crystal.  So  the  biaxial  crystals  considered  up  to  the  present 
have  three  planes  of  optical  symmetry.  From  Fig.  86  one  sees 
directly  that  in  the  case  of  such  a  crystal  the  vividness  of  the 
colors  and  their  sequence  from  within  outward  must  be  the 
same  with  both  hyperbolas,  because  the  bisectrices  for  all 
colors  absolutely  coincide  at  the  center  C.  When  this  latter  is  not 
the  case,  however,  the  crystal  having  in  addition  a  dispersion  of  the 
principal  vibration  directions,  then  the  color  edgings  of  the  two 
hyperbolas  can  no  longer  be  exactly  alike;  and  this  unlikeness 
itself  presents  the  most  delicate  means  of  recognizing  such  a  dis- 
persion, —  i.e.  the  existence  of  a  lower  grade  of  optical  symmetry. 

The  difference  in  the  orientation  of  the  principal  vibration 

*  In  consequence  of  two  principal  refractive  indices  being  equal  for  this  color 
the  index-surface  for  it  is  a  rotation  ellipsoid.  Here,  therefore,  arises  the  special 
case  mentioned  on  p.  126,  but  only  for  the  light  of  one  definite  vibration  period, 
while  in  the  case  of  really  optically  uniaxial  crystals  the  index-surface  is  a  spheroid 
for  all  colors. 


174  OPTICALLY   BIAXIAL   CRYSTALS 

directions  for  different  colors  may  be  either  complete  or  par- 
tial. That  is  to  say,  one  of  these  vibration  directions  may  be 
for  all  colors  the  same,  while  the  other  two  are  dispersed  in 
the  plane  standing  perpendicular  to  it  (this  plane  is  then  a 
common  principal  optic  section  for  all  colors,  and  the  crystal 
has  only  this  one  plane  of  optical  symmetry);  or  else  all  three 
principal  vibration  directions  may  vary  their  crystallographic 
orientation  with  the  vibration  period  of  the  light  (the  crystal 
then  having  no  plane  of  optical  symmetry)  *  In  the  former  of 
these  two  cases,  as  will  be  seen  from  the  following,  the  inter- 
ference phenomena  assume  different  forms,  according  as  the 
common  principal  vibration  direction  for  all  colors  is  that  of  the 
intermediate  light  velocity  or  of  one  of  the  other  two.| 

When  the  vibration  direction  of  the  light  rays  having  the 
intermediate  refractive  index  /?  has  the  same  crystallographic 
orientation  for  all  colors,  this  holds  true  also  of  the  plane  per- 
pendicular to  that  direction;  i.e.  of  the  principal  section  XZ  of 
the  ray-surface  (Fig.  64).  The  optic  axes  for  all  colors  then 
lie  in  one  and  the  same  plane,  but  their  bisectors  are  dispersed. 
Hence,  if  from  such  a  crystal  a  plate  is  made  that  exhibits  in 
convergent  light  the  interference-figure  of  both  axes,  such  a 
plate,  although  it  may  indeed  be  cut  perpendicular  to  the  axial 
plane  for  all  colors,  can  be  normal  to  the  acute  bisectrix  only 
for  one  single  color;  e.g.  for  an  intermediate  color  lying  in  the 
brightest  part  of  the  spectrum.  Then  does  the  acute  bisectrix 
for  the  colors  of  lesser  refrangibility  stand  inclined  to  the  plane 

*  The  coinciding  of  two  principal  vibration  directions  for  all  colors  leads 
naturally  to  the  coincidence  of  the  third,  perpendicular  to  those  two;  wherefore, 
besides  the  cases  treated  in  the  last  section,  only  the  two  mentioned  above  are 
possible. 

t  For  illustrating  the  optical  relations  that  result  from  this,  service  may  well 
be  made  either  of  the  dispersion  models  supplied  by  Bohm  and  Wiedemann  of 
Munich  (see  Appendix),  in  which  the  ray-surface  for  each  of  three  different  colors 
is  represented  in  a  manner  similar  to  that  mentioned  on  p.  135;  or  of  the  little 
glass  models  with  threads  stretched  through  which  are  made  for  a  small  price 
(five  or  six  marks)  by  C.  Mohn,  an  employee  (Diener)  in  the  Mineralogical 
Institute  of  Rothstock  University  in  Mecklenburg. 


INTERFERENCE   PHENOMENA   IN   CONVERGENT   LIGHT     175 

of  the  plate  in  one  direction,  the  bisectrix  for  those  of  greater 
refrangibility  in  the  other  direction;  and  since  the  rays  parallel 
to  the  several  bisectrices  must,  on  emerging,  still  suffer  a  refrac- 
tion, their  divergence  is,  in  appearance,  still  increased.  With 
the  majority  of  crystallized  substances  this  divergence  amounts 
moreover  only  to  one  or  two  degrees,  frequently  still  less,  seldom 
considerably  more.  If  the  point  in  the  conoscope  at  which  all 
the  rays  converge  that  traverse  the  plate  perpendicularly,  i.e. 
the  center  of  the  field,  be  C  (Fig.  87),  and  if  the  direction  of 


these  rays  may  correspond  to  the  acute  bisectrix  for  an  inter- 
mediate color,  then  is  the  bisectrix  for  red  inclined  in  one  direc- 
tion and  that  for  violet  in  the  other  direction,  but  all. three  lie  in 
the  plane  indicated  by  the  straight  line  55.  Hence,  if  R  be  the 
point  in  the  field  where  all  the  rays  converge  that  pass  through 
the  plate  parallel  to  the  acute  bisectrix  for  red,  and  r,r  the  axial 
points  for  the  same  color,  then  the  lemniscates  drawn  full  rep- 
resent those  that  appear  when  the  instrument  is  illuminated 
with  light  of  this  color.  Similarly,  if  V  is  the  point  of  con- 
vergence of  the  rays  that  pass  through  the  crystal  parallel  to 
the  acute  bisectrix  for  violet,  and  v,v  the  axial  points  for  this 
same  color,  whose  axial  angle  is  of  course  different  (in  the 
figure  p>o  is  assumed),  then  are  the  dotted  lemniscates  those 
appearing  in  homogeneous  violet  light.  If,  then,  we  observe  the 


176  OPTICALLY   BIAXIAL   CRYSTALS 

interference-figure  in  white  light,  there  follows  from  Fig.  87 
directly  that,  although  this  figure  must  be  symmetrical  to  the 
line  55,  and  thus  identical  above  and  below,  yet  in  no  case  will 
it  be  symmetrical  to  the  line  MM.  While  the  axial  figure  of 
a  crystal  having  parallel  principal  vibration  directions  for  all 
colors  (cf.  Fig.  85,  p.  169)  is  symmetrical  right  and  left  also, 
the  latter  identity  here  comes  to  an  end,  in  consequence  of  the 
dispersion  of  the  bisectrices:  on  the  right  and  on  the  left  the  curves 
overlap  one  another  in  different  ways,  and  so  the  color  distri- 
bution in  the  right-  and  in  the  left-hand  rings  cannot  be  the  same; 
consequently  the  rings  of  the  two  systems  will  be  of  different 
size  and  colored  with  different  vividness,  and  in  the  two  systems 
the  sequence  of  the  colors  will  be  different;  all  these  differences 
are  the  greater,  the  greater  the  dispersion  of  the  bisectrices. 
When  the  crystal  is  turned  so  that  the  optic  axial  plane  is  parallel 
to  the  vibration  direction  of  one  of  the  two  nicols,  the  black 
bar  appears  parallel  to  55,  as  with  a  crystal  of  the  first  kind; 
and  without  color  edging,  since  the  optic  axial  planes  for  all 
colors  coincide.  The  interference  phenomenon  exhibited  by  a 
plate  of  gypsum  in  this  position  is  portrayed  in  Fig.  6  of  Plate 
II;  and  here  the  difference  that  stands  out  between  the  two 
systems  is  to  be  distinctly  perceived,  especially  for  the  inner- 
most rings.  If  one  rotates  the  plate  45°,  so  that  the  dark  hyper- 
bolas appear,  the  difference  stands  out  still  more;  for,  since  on 
account  of  the  dispersion  of  the  bisectrices  the  points  r  and  v 
(Fig.  87)  do  not  stand  at  the  same  distance  from  the  center 
on  the  right  as  on  the  left,  the  color  edging  exhibited  by  the  two 
hyperbolas  on  their  two  sides  —  and  this  edging,  as  we  know,  is 
most  distinct  within  the  first  ring  —  must,  with  the  two  hyper- 
bolas, be  of  different  width  and  different  vividness.  If  the  dis- 
persion of  the  bisectrices  is  great  and  that  of  the  axes  likewise 
great  (i.e.  the  axial  angle  very  different  for  red  and  for  violet), 
which  case  is  realized  in  Fig.  87,  then  on  one  side  of  the  center 
the  axial  point  r  for  red  lies  inward,  v  that  for  violet  outward,  and 
on  the  other  side  r  outward,  v  inward;  since,  further,  the  distance 


INTERFERENCE  PHENOMENA   IN   CONVERGENT   LIGHT      177 

of  both  r  and  v  from  the  center  is  very  different  on  the  right 
and  on  the  left,  the  left  hyperbola  would  appear  edged  inwardly 
with  red,  outwardly  with  blue,  and  very  broadly,  while  the  right 
would  exhibit  only  narrow  color  edgings,  the  inner  one  blue  and 
the  outer  red.  Were  the  dispersion  of  the  bisectrices  some- 
what less,  then  on  one  side  the  axial  points  for  the  different 
colors  would  almost  entirely  coincide;  in  the  interference- 
figure  there  would  then  be  seen,  instead  of  two  reversely  colored 
hyperbolas,  one  with  distinct  color  edgings  and  the  other  with- 
out them.  With  still  less  dispersion,  however,  — as  is  more 
frequently  met  with, —  although  the  distance  at  which  r  and  v 
stand  from  the  center  is  indeed  different  on  the  right  and  on  the 
left,  yet  these  points  do  not  lie  reversed,  but  either  with  both  v's 
outward  or  with  both  toward  the  center.  Then  do  both  hyper- 
bolas appear  edged  inwardly  with  red  and  outwardly  with  blue, 
or  vice  versa,  but  with  different  color  shade  and  different  vivid- 
ness of  coloring.  This  last  case  is  represented  by  Fig.  7  of 
Plate  II,  which  refers  to  the  same  gypsum  plate  as  Fig.  6,  after  a 
rotation  of  45°.  One  should  note  herewith,  besides  the  differ- 
ence between  the  two  hyperbola  edgings,  more  especially  that 
between  the  inner  side  of  the  first  ring  on  the  right  and  on  the 
left.  Since  this  kind  of  dispersion  of  the  colors  of  the  axial 
figure  is  due  to  an  unequal  inclination  of  the  principal  vibra- 
tion directions  within  the  same  plane,  it  is  called  inclined  dis- 
persion (Des  Cloiseaux's  "  dispersion  inclinee"). 

When  on  the  other  hand  the  principal  vibration  direction  com- 
mon to  all  colors  is  that  of  the  greatest  or  of  the  least  light  velocity, 
i.e.  one  of  the  two  mean  lines  of  the  optic  axes,  it  is  clear  that  the 
plane  of  the  latter  must  now  have  for  different  colors  a  different 
orientation.  For  if  the  bisectors  of  the  obtuse  axial  angle  for  all 
colors  have  the  same  direction,  those  of  the  acute  angle,  and  con- 
sequently also  the  axial  planes  of  the  different  colors,  are,  as  it 
were,  rotated  about  that  direction;  while  if  the  bisectrix  coinciding 
for  all  colors  is  the  acute,  the  bisectors  of  the  obtuse  angle,  and 
therefore  also  the  axial  planes,  diverge  about  the  acute  bisectrix. 


I78 


OPTICALLY   BIAXIAL   CRYSTALS 


If  from  a  crystal  corresponding  to  the  first  of  the  two  cases 
just  mentioned  we  cut  a  plate  perpendicular  to  the  acute  bisec- 
trix for  an  intermediate  color,  the  optic  axial  planes  for  less 
refrangible  light  rays  are  oblique  to  the  plate  in  one  direction, 
those  for  more  refrangible  in  the  other  direction.  But  the 
bisectrices  for  all  colors  lie  in  a  plane  perpendicular  to  the  plate ; 
namely,  in  that  principal  section  that  stands  perpendicular  to 
the  common  principal  vibration  direction  (the  obtuse  bisectrix). 
Let  this  plane  (the  plane  of  optical  symmetry)  be  indicated  in 
Fig.  88  by  the  straight  line  SS,  and  let  C  again,  as  in  the  last 


figure,  be  the  center  of  the  field  in  the  conoscope,  i.e.  the  point 
of  convergence  of  all  rays  that  pass  through  the  crystal  plate 
perpendicularly;  then  the  direction  of  these  rays  is  identical  with 
the  bisectrix  only  for  an  intermediate  wave  length,  while  for 
other  colors  the  rays  lie  inclined  in  part  upward,  in  part  down- 
ward, in  the  plane  SS.  Hence,  if  R  be  the  point  where  all  rays 
converge  that  in  the  crystal  move  parallel  to  the  bisectrix  for 
red,  and  if  ryr  be  the  axial  points  for  this  color,  then  the  line  rr  con- 
necting these  two  points  must  be  both  normal  to  SS  and  bisected 
by  it,  since  MM  is  the  obtuse  bisectrix.  For  violet  the  point  at 
which  the  rays  parallel  to  the  bisectrix  converge  must  lie  on  the 


INTERFERENCE   PHENOMENA   IN   CONVERGENT   LIGHT     179 

other  side  of  C,  —  about  at  V',  and  the  axial  points  v,v  like- 
wise must  lie  symmetrically  on  the  right  and  left  of  55,  but  at 
another  distance,  since  the  axial  angle  for  this  color  is  other  than 
for  red.  From  Fig.  88,  then,  it  is  easy  to  understand  how  the 
interference-figure  must  look  in  white  light;  it  will  be  absolutely 
symmetrical  with  reference  to  the  line  55,  because  the  right  and 
left  sides  will  have  entirely  equal  and  opposite  color  distri- 
bution. Not  so  the  upper  and  lower  halves,  since  the  rings  for 
the  different  colors  do  not  lie  equally  on  the  two  sides  of  MM; 
at  their  upper  and  lower  sides  the  color  rings  will  thus  exhibit 
different  colors.  When  the  optic  axial  planes  of  the  crystal  are 
parallel  to  the  principal  section  of  one  of  the  two  crossed  nicols 
of  the  apparatus,  their  difference  in  position  for  the  different 
colors  will  manifest  itself  most  distinctly  at  the  dark  middle  bar, 
which  gives  the  position  of  the  axial  plane  and  by  reason  of  the 
variation  of  the  same  for  the  different  colors  exhibits  a  colored 
edging  above  and  below;  either  blue  above  (because  the  black 
bar  for  red  light  lies  there)  and  red  below,  as  in  the  case  rep- 
resented in  Fig.  88,  or  vice  versa.  This  dispersion,  because 
with  it  the  color  distribution  varies  on  the  different  horizontal 
lines,  has  been  named  horizontal  dispersion  (Des  Cloiseaux's 
"  dispersion  horizontale").  Figure  8  of  Plate  II  represents  the 
interference-figure  of  a  feldspar  (sanidine)  crystal  in  the  position 
where  its  axial  planes  are  parallel  to  the  principal  section  of  a  nicol, 
in  which  position  therefore  the  colored  edgings  are  to  be  seen; 
Fig.  9  shows  the  interference-figure  of  the  same  plate  when  its 
axial  plane  forms  45°  with  the  nicols.  In  the  latter  figure  the 
difference  between  the  color  curves  on  the  two  sides  of  the  line 
connecting  the  axial  points  can  be  seen  even  better  than  in  Fig.  8. 
The  second  of  the  two  cases  mentioned  on  page  177  is  that 
in  which  the  acute  bisectrix  is  identical  for  all  colors.  A  plate 
cut  perpendicular  to  this  direction  stands  normal  at  once  to  the 
optic  axial  planes  for  all  colors;  but,  since  these  planes  are 
rotated  about  the  common  principal  vibration  direction  (i.e. 
the  normal  to  the  plate) ,  they  intersect  the  plane  of  the  plate  in 


OPTICALLY  BIAXIAL   CRYSTALS 

different  directions.  Let  C  (Fig.  89)  again  be  the  center  of  the 
field,  corresponding  to  the  acute  bisectrix  for  all  colors;  if 
further  r,r  be  the  axial  points  for  red  and  the  curves  drawn  full 
the  lemniscates  for  this  color,  then  does  the  straight  line  rr 
indicate  the  position  of  the  axial  plane  for  red.  For  another 
color,  as  violet,  although  the  acute  bisectrix  is  indeed  the  same, 
yet  in  the  principal  section  perpendicular  to  it  the  obtuse 


Fig.  89. 

takes  another  position;  consequently  the  axial  plane  for  this 
color  is  different,  being  rotated  about  C  by  a  certain  angle. 
Let  its  position  be  represented  by  the  straight  line  w;  and  for 
the  same  color  let  v,v  be  the  two  axial  points,  the  dotted  curves 
the  lemniscates.  The  axial  planes  for  the  intermediate  colors 
naturally  lie  between  rr  and  w.  Hence  it  follows  that  only 
in  monochromatic  light  can  the  axial  figure  exhibit  symmetry 
with  reference  to  two  mutually  perpendicular  straight  lines;  in 
white  light,  on  the  other  hand,  with  reference  to  none:  rather 
is  the  color  distribution  different  on  the  right  and  on  the  left,  like- 
wise above  and  below;  but  it  must  be  identical  on  the  right 
below  and  on  the  left  above,  as  well  as  on  the  right  above  and 
on  the  left  below.  In  other  words,  the  interference-figure  must 
be  the  same  in  all  respects  when  it  is  rotated  180°  about  the 
normal  to  the  plate  at  C.  Thus  it  exhibits  symmetry  only  with 
reference  to  the  center,  which  may  therefore  be  designated  as 
the  "  center  of  symmetry  "  of  the  figure.  Since  the  axial  planes 
for  the  different  colors  are  all  rotated  about  C,  the  black  middle 
bar  appearing  in  the  case  of  parallel  position  with  a  nicol  must 
in  this  case  too  be  edged  with  color,  but  differently  on  the  right 
and  on  the  left;  so  either  red  on  the  right  above  and  on  the 


INTERFERENCE    PHENOMENA  IN  CONVERGENT   LIGHT      l8l 

left  below,  and  blue  on  the  left  above  and  on  the  right  below, 
or  vice  versa.  It  is  by  this  cross-wise  agreement  of  the  color- 
ing of  the  middle  bar,  and  of  the  inner  rings,  that  this  disper- 
sion, which  is  known  as  crossed  dispersion  (Des  Cloiseaux's 
"  dispersion  croisee  ou  tournante  ")  is  most  easily  recognized. 
The  interference  phenomenon  produced  by  a  plate  of  borax  in 
this  position  is  shown  in  Fig.  10  of  Plate  II,  while  Fig.  n  repre- 
sents the  one  exhibited  after  the  plate  has  been  rotated  45°. 

Were  the  optic  axial  angle  of  a  crystal  so  little  different 
from  90°,  and  its  refractive  indices  so  small,  that  the  axial 
figures  could  be  seen  not  only  through  a  plate  perpendicular  to 
the  acute,  but  also  through  one  perpendicular  to  the  obtuse 
bisectrix,  then  it  is  clear  that  one  of  these  plates  would  ex- 
hibit the  phenomenon  of  horizontal,  the  other  that  of  crossed 
dispersion,  — always  supposing  their  axial  planes  to  stand  perpen- 
dicular to  the  common  principal  section  for  all  colors.  In  reality, 
therefore,  the  two  last-mentioned  kinds  of  dispersion  are  in  such 
crystal  always  combined,  except  that  usually  only  one  of  them  can 
be  observed,  because  of  the  obtuse  axial  angle's  being  too  large. 

The  crystal's  considered  on  pages  174-181  may  accordingly  be 
designated  as  those  whose  optical  phenomena  exhibit  symmetry 
with  respect  to  a  plane  and  to  a  "  binary  axis"  normal  to  that 
plane,  —  a  "  binary  axis  "  being  a  line  such  that  the  crystal,  when 
rotated  about  it,  exhibits  identical  behavior  in  two  positions 
differing  from  each  other  by  a  rotation  of  180°. 

There  yet  remains,  finally,  only  that  case  to  consider  in 
which  all  three  principal  vibration  directions  for  the  different 
colors  of  the  spectrum  have  a  different  orientation  in  the  crystal. 
Here,  therefore,  when  we  pass  over  from  one  color  to  another, 
there  is  a  change  not  only  in  the  form  of  the  index-surface,  — 
this  being  defined  by  the  ratios  among  the  three  principal  refrac- 
tive indices,  —  but  also  in  the  directions  of  its  three  mutually 
perpendicular  axes.  If  from  such  a  crystal  a  plate  is  cut  per- 
pendicular to  the  acute  bisectrix  for  an  intermediate  color,  as 
yellow,  then  for  every  other  color  its  plane  is  oblique  not  only  to 


1 82  OPTICALLY  BIAXIAL  CRYSTALS 

the  bisector  of  the  optic  axes,  but  also  to  their  plane.  In  Fig. 
90  therefore,  if  C  is  again  the  center  of  the  field,  and  if  y,y  are 
the  points  where  the  rays  corresponding  to  the  two  optic  axes 
for  yellow  converge  in  the  field  of  view,  the  plane  of  the  optic 
axes  for  yellow  thus  intersecting  that  of  the  plate  in  the  line  yy, 
then  the  points  and  lines  that  correspond  respectively  to  the 
bisectrices  and  axial  planes  for  the  remaining  colors  are  all 
other  than  these.  For  example,  let  R  be  the  point  in  the  field 
where  those  rays  converge  that  passed  through  the  crystal 
parallel  to  the  acute  bisectrix  for  red,  and  let  r,r  be  the  two  axial 
points  for  this  color,  while  F,  v,  and  v  represent  the  analogous 


Fig.  90. 

points  for  violet.  From  the  lemniscates  drawn  here,  for  these 
colors,  in  the  same  way  as  in  the  preceding  figures,  it  is  at  once 
understood  that  the  arising  interference-figure  must  have  a 
color  distribution  that  is  wholly  unsymmetrical ;  because  neither 
right  and  left  nor  above  and  below  can  there  be  any  identity, 
while  just  as  little  can  the  figure  be  the  same  after  a  half  rota- 
tion about  its  center.  As  an  interference-figure  corresponding 
to  this  relative  position  of  the  principal  vibration  directions, 
that  of  copper  sulphate,  a  bluish-colored  salt,  is  shown  in  Fig's 
12  and  13  on  Plate  II. 

In  case  the  angles  between  the  optic  axial  planes  for  differ- 
ent colors  are  greater  than  with  this  salt,  then,  in  white  light, 
color  curves  of  lemniscate-like  shape  are  formed  no  longer, 
but  an  interference-figure  arises  similar  to  the  one  represented 
in  Fig.  5,  Plate  II;  differing  from  that  figure,  however,  by  the 
circumstance  that  the  color  distribution  varies  in  all  directions. 


DETERMINATION   OF  OPTIC  AXES  AND   THEIR  ANGLE     183 

The  dispersion  phenomena  described  in  the  foregoing,  which 
offer  such  a  sensitive  means  of  recognizing  an  unlike  orienta- 
tion of  the  principal  vibration  directions  in  a  crystal,  are  of 
special  practical  importance  for  this  reason:  the  coincidence 
or  divergence  of  these  directions  for  the  different  colors  stands 
in  a  regular  relation  to  the  symmetry  of  the  form  of  the  crys- 
tals in  question;  so  that  from  the  interference-figure  observed 
in  a  particular  case,  or  from  the  manner  of  the  color  distri- 
bution in  it,  a  conclusion  may  be  drawn  as  to  the  symmetry  of 
the  crystal  investigated. 

DETERMINATION  OF  THE  OPTIC  AXES  IN  BIAXIAL  CRYSTALS 
AND  MEASUREMENT  OF  THEIR  ANGLE 

Now  the  phenomena  considered  in  the  foregoing  section,  the 
interference-figures  of  biaxial  crystals,  to  be  observed  in  the 
conoscope,  present  the  means  of  finding  the  position  of  the  two 
optic  axes  and  thereby  of  ascertaining  the  crystallographic 
orientation  of  the  principal  vibration  directions,  which,  accord- 
ing to  page  151,  is  requisite  for  the  determination  of  the  optical 
constants  of  the  crystal.  For,  if  a  plane  face  of  the  crystal 
gives  egress  to  rays  transmitted  parallel  to  an  optic  axis,  i.e. 
if  in  the  field  of  the  conoscope  we  see  the  ring  system  inter- 
sected by  one  dark  bar,  which  surrounds  the  axial  point,  we 
are  able,  by  methods  to  be  discussed  farther  on,  to  measure  the 
angle  the  optic  axis  in  question  forms  with  the  normal  to  this 
face.  If  in  addition  the  same  rays  are  caused  to  emerge 
through  another  crystal  face,  whose  angle  with  the  former  one 
is  known,  and  if  we  here  too  determine  their  inclination  to  the 
normal  to  the  face  of  emergence,  then  from  these  data  the  crys- 
tallographic orientation  of  the  optic  axis  may  be  calculated. 
When  the  same  procedure  is  feasible  also  for  the  second  optic 
axis,  the  orientation  is  thereby  given  of  the  three  principal  vibra- 
tion directions  — of  the  acute  and  the  obtuse  bisectrix  and  the 
normal  to  the  optic  axial  plane.  On  account  of  the  dispersion 
of  these  directions  the  measurements  in  question  must  of  course 


184  OPTICALLY   BIAXIAL   CRYSTALS 

be  carried  out  specially  for  each  color.  The  problem  assumes 
its  simplest  form  when,  as  occurs  with  many  biaxial  crystals 
having  certain  kinds  of  symmetry,  one  natural  face  of  the 
crystal  exhibits  the  two  axial  figures  symmetrically  in  the  field 
of  the  conoscope.  For  then  the  normal  to  the  plate  is  the  acute 
bisectrix,  and  the  line  parallel  to  the  plate  and  connecting  the 
two  axial  points  the  obtuse  bisectrix,  while  the  normal  common 
to  these  two  directions  is  the  vibration  direction  of  intermediate 
light  velocity. 

Supposing  that  in  this  way  the  crystallographic  orientation  of 
the  three  principal  vibration  directions  has  been  found,  the 
three  principal  refractive  indices  can  now  be  determined  by  one 
of  the  methods  given  on  pages  144-150.  From  these  indices  one 
may  calculate  the  optic  axial  angle,  according  to  the  formula 
given  on  page  143;  the  same  can  also  be  determined  directly, 
however,  by  methods  that,  on  the  exit  from  the  crystal  of  the 
rays  corresponding  to  the  optic  axes,  prevent  these  rays  suffer- 
ing a  deviation.  This  would  of  course  also  be  the  case  if  the 
plane  at  which  emerge  the  rays  that  correspond  to  the  first  axis 
were  exactly  perpendicular  to  that  axis,  and  the  face  of  emer- 
gence of  the  rays  passing  through  parallel  to  the  second  axis, 
exactly  perpendicular  to  this  axis.  But  without  knowing  the 
axial  angle,  the  determination  of  which  is  primarily  the  object  of 
the  method,  such  plane  surfaces  cannot  be  produced.  On  the 
other  hand,  the  rays  in  question  emerge  wholly  undeviated, 
from  the  crystal,  if  this  is  ground  down  to  the  form  of  a 
sphere,  or  of  a  cylinder  whose  axis  is  perpendicular  to  the  optic 
axial  plane.  Those  rays,  A  A  and  A' A'  (Fig.  91),  that  pass 
exactly  through  the  center  of  the  sphere  or  cylinder,  and  par- 
allel to  the  two  axes,  always  strike  the  surface  at  a  point  where 
the  surface  is  perpendicular  to  the  axis,  and  consequently  are 
not  refracted.  If,  therefore,  the  sphere  or  cylinder  were  rotat- 
able  about  an  axis  passing  through  its  center  and  perpendic- 
ular to  the  plane  of  Fig.  91,  then  by  adjusting  the  axial  figure 
of  A  and  that  of  A'  in  a  fixed  polarization  apparatus  we 


DETERMINATION  OF  OPTIC  AXES  AND   THEIR  ANGLE       185 

could  measure  the  angle  AC  A1 ',  in  virtue  of  the  rotation  requisite 
for  these  adjustments.  But,  owing  to  the  difficulty  of  preparing 
so  perfect  a  sphere  or  such  a  cylinder, 
it  is  preferable  to  determine  the  axial 
angle  in  some  other  way;  e.g.  by  immers- 
ing the  crystal  in  a  liquid  whose  refractive 
index  is  approximately  equal  to  /?  of  the 
crystal;  for  then,  at  the  boundary  of 
crystal  and  liquid,  no  deviation  of  the 
light  rays  occurs,  wherefore  the  crystal 
may  be  of  any  shape.  On  this  principle 
is  based  the  rotation  apparatus  con- 
structed by  C.  Klein,*  by  means  of 
which,  even  when  the  refringency  of  a 

crystallized  substance  is  known  only  approximately,  one  is 
able  with  a  crystal,  or  a  fragment  of  one,  of  any  shape  to 
make  an  approximate  determination  of  the  orientation  of  the 
two  optic  axes. 

For  exact  determination  of  the  optic  axial  angle,  service  is 
made  of  a  plane-parallel  plate  cut  perpendicular  to  the  bisector 
of  the  acute  axial  angle.  When  it  is  a  matter  of  a  crystal  with- 
out dispersion  of  the  principal  vibration  directions,  the  plate 
mentioned  is  perpendicular  to  the  acute  bisectrix  for  all  colors; 
otherwise,  only  for  a  definite  one.  Yet,  since  the  dispersion  of 
the  bisectrices  is  in  most  cases  but  slight,  a  plate  cut  normal  to 
the  bisectrix  for  yellow  commonly  stands  very  nearly  perpen- 
dicular to  that  for  the  remaining  colors  as  well,  and  may 
therefore,  unless  the  greatest  accuracy  is  required,  serve  also  to 
determine  the  axial  angle  for  red,  blue,  etc. 

In  Fig.  92  (next  page)  let  PPP'P'  be  the  intersection  of  such 
a  plate  with  the  plane  of  the  optic  axes,  and  let  the  normal  to  the 
plate,  MM',  be  the  bisector  of  their  acute  angle;  then  the  ray  sys- 

*  Described  and  illustrated  by  figures  in  Phys.  Kryst.  4th  ed.  782-783;  3rd  ed. 
749-750.  See  also,  Sitzungsber.  d.  Akad.  d.  Wissensch.  Berlin,  1895,  9r>  Leiss: 
Die  op.  Instrum.,  232. 


i86 


OPTICALLY   BIAXIAL   CRYSTALS 


terns  that  are  parallel  to  the  two  axes  will  strike  the  surface  at  equal 
but  opposite  angles  and  therefore  suffer  an  equal  refraction  in 

the  opposite  direction.  While 
these  ray  systems  form  within 
the  crystal  the  angle  AC  A',  the 
true  optic  axial  angle,  they  in- 
clude after  their  exit  a  greater 
one,  the  so-called  apparent  axial 
angle,  BDB'\  and  this  angle, 
precisely  as  /.ACA' ',  is  bisected 
by  MN'.  The  apparent  axial 
angle  may  be  measured  in  the 
following  way:  — 

One    inserts    the    plate    (let 
PPP'P',  Fig.  93,  be  its  section, 

as  above)   between  condensing  lens  and  objective  of  the  cono- 
scope  in  such  a  way  that  it  is  rotatable  about  an  axis  passing 


Fig.  93.  Fig.  94.  Fi8-  95- 

exactly  perpendicular  to  the  optic  axial  plane  and  approxi- 
mately through  the  center  of  the  plate.  In  Fig.  93  are  given,  in 
vertical  section,  the  near-by  parts  only,  of  the  instrument;  and 
the  plate  is  in  the  position  where  it  exhibits  the  interference- 
figure  symmetrical,  since  the  acute  bisectrix  coincides  with  the 


DETERMINATION  OF  OPTIC  AXES  AND  THEIR  ANGLE     187 

axis  of  the  instrument.  Now  the  above-mentioned  rotation  is 
realized  by  fastening  rigidly  to  the  instrument,  above  the  plane 
of  the  figure,  a  graduated  circle  through  whose  center  passes  a 
rotatable  axis;  this  axis  stands  normal  to  the  plane  of  the  figure 
and,  terminating  in  a  pincette,  carries  the  plate.  From  the  per- 
spective view  in  Fig.  94  the  arrangement  of  this  apparatus  and 
the  possibility  of  measuring  a  rotation  of  the  plate  with  its  aid, 
will  be  understood  directly.  Supposing,  then,  the  axis,  and 
thus  the  crystal  plate,  to  be  rotated  until  the  rays  (aa  in  Fig.  95), 
that  traversed  it  along  an  optic  axis  enter  the  objective  exactly 
parallel  to  the  axis  of  the  instrument,  these  rays  will  converge 
at  the  center  of  the  field;  consequently  the  center  of  one  of  the 
ring  systems  will  appear  exactly  at  the  center  of  the  field.  This 
point  may  be  marked  by  a  pair  of  cross-hairs  in  the  focal 
plane  of  the  conoscope;  and  the  adjustment  of  the  axial  figure 
on  the  intersection  of  the  cross-hairs  is  accomplished  with 
special  accuracy  if  we  employ  the  interference-figure  with  the 
hyperbolas,  — i.e.  if  we  cause  the  crossed  nicols  of  the  instru- 
ment to  form  45°  with  the  axial  plane  of  the  plate.  Since  always 
the  adjustment  of  an  optic  axis 
can  be  made  only  for  one  definite 
color,  the  apparatus  must  of  course 
be  illuminated  with  homogeneous 
light,  —  a  sodium  flame,  for  ex- 
ample. The  interference  figure  c 
then  appears  as  portrayed  in  Fig. 
96,  where  CC  and  C'C'  represent 
the  cross-hairs  of  the  telescope, 
NN  and  N'N'  the  vibration  direc- 
tions  of  the  two  nicols.  When 
the  plate  has  been  rotated  until 
the  middle  of  the  dark  hyperbola  and  the  vertical  cross-hair 
absolutely  coincide,  as  shown  in  the  figure,  it  has  exactly  the 
position  indicated  in  Fig.  95.  So  if  we  rotate  back  to  the 
original  position  and  just  as  far  in  the  other  direction,  until  the 


i88 


OPTICALLY   BIAXIAL   CRYSTALS 


second  axial  figure  is  in  the  middle  of  the  field  in  exactly  the 
same  way,  i.e.  until  a! a'  of  Fig.  95  coincides  with  the  axis  of 
the  instrument,  we  must  obviously,  between  these  two  adjust- 
ments of  the  one  and  the  other  optic  axis  respectively,  on  the 
center,  have  had  to  rotate  to  the  amount  of  the  axial  angle 
after  exit  into  air.  Therefore  the  rotation  to  be  read  off  on  the 
circle  gives  directly  the  apparent  axial  angle  for  the  color  used. 
If  the  instrument  is  illuminated  with  light  of  another  color,  we 
obtain,  because  of  the  dispersion  of  the  axes,  other  readings 
for  both  adjustments  and  thus  a  larger  or  a  smaller  axial  angle. 
There  has  now  to  be  determined  the  relation  in  which  the 

apparent  axial  angle  stands  to 
the  true.  If  PPP'P'  (Fig.  97) 
again  be  the  intersection  of  the 
crystal  plate  with  the  axial  plane, 
MM'  the  acute  bisectrix  and  at 
the  same  time  the  normal  to  the 
plate,  and  A  A'  an  optic  axis,  then 
obviously  Va  =  A'AMr  is  half  the 
true  axial  angle  and  E  =  BAM 
is  half  the  apparent,  so  that  the 
true  (interior)  and  the  appar- 
ent (exterior)  angles  are  respec- 
tively 2  Fa  and  2.E.  Now  according  to  page  155  a  ray  trans- 
mitted along  the  direction  A  A'  has  the  intermediate  light 
velocity;  it  is  therefore  at  the  point  A  so  refracted  that  its  re- 
fractive index  from  air  into  the  crystal  is  the  intermediate  prin- 
cipal refractive  index  /?.  When  the  ray  AA'  emerges,  refracted, 
into  the  air,  its  angle  of  incidence  is  Va  and  its  angle  of  refrac- 
tion is  .E;  obviously,  therefore, 


sin£ 
sin  V, 


=  /?,  and  consequently  sin  E  =  /?  •  sin  Va. 


By  means  of  this  equation  we  determine  the  ratio  between  the 
true  axial  angle  and  the  apparent  axial  angle  in  air.     If,  therefore, 


DETERMINATION  OF  OPTIC  AXES  AND   THEIR  ANGLE      189 

one  has  measured  all  three  principal  refractive  indices  for  a 
definite  color  and  from  them  deduced  the  true  axial  angle  for 
the  same  color,  the  apparent  follows  from  the  above  equation. 
Hence,  if  the  apparent  angle  is  determined  directly,  in  the 
manner  described,  a  comparison  of  it  with  the  values  calculated 
from  the  refractive  indices,  only,  supplies  a  standard  fpr  judg- 
ing of  the  accuracy  with  which  the  latter  have  been  determined ; 
and  this  so  much  the  more,  as  in  most  cases  — namely,  when  for 
the  preparation  of  prisms  and  plates  one  has  only  small  crystals 
at  one's  disposal  — the  measurement  of  the  axial  angle  is  more 
accurate  than  of  the  refractive  indices.  But  still  more  impor- 
tant is  the  determination  of  the  apparent  axial  angle  in  those 
cases  where  the  development  of  the  crystal  permits  only  in  one 
direction  the  preparation  of  prisms  large  enough  for  accurate 
determination  of  the  refractive  indices,  in  consequence  whereof 
only  two,  at  the  most,  of  the  principal  refraction  quotients  can 
be  determined.  Provided  these  two  are  not  a.  and  7-,  but  either 
a.  and  /?  or  /?  and  /-,  then  with  the  aid  of  f)  we  may  from  the 
apparent  axial  angle  calculate  the  true;  and  from  this  angle  and 
the  two  measured  refractive  indices  the  third  index  may  be  found 
by  solving  for  it  in  the  equation  given  on  page  143. 

If  the  angle  of  the  optic  axes  exceeds  a  certain  size,  the  rays 
parallel  to  the  axes  can  no  longer  emerge  into  air.     For  when 

Va  is  such  that  sin  Va  =  — ,  then  sin  E  is  unity  and  the  apparent 

axial  angle  therefore  180°;  so  that,  beginning  with  this  value 
for  Va,  a  value  which  depends  on  the  intermediate  refractive 
index  of  light  passing  from  the  crystal  into  air,  total  reflection 
of  those  rays  occurs.  Were  the  crystal  to  be  surrounded,  in- 
stead of  with  air,  with  a  fluid  in  which  the  light  velocity  differed 
less  from  that  in  the  crystal,  then  at  the  boundary-surface  of  the 
two  media  the  rays  corresponding  to  the  optic  axes  would  be 
less  deviated;  and  if  in  this  fluid  the  transmission  velocity  of 
light  were  even  less  than  in  the  crystal  itself,  these  rays  would 
be  refracted  toward  the  axis  of  incidence.  That  is  to  say,  in 


190 


OPTICALLY   BIAXIAL   CRYSTALS 


this  medium  the  apparent  optic  axes  would  include  an  angle 
smaller  than  the  true.     Let  PPP'P'   (Fig.  98)    be  the   crystal 

plate  and  HH  the  boundary-sur- 
face, parallel  to  the  plate,  between 
the  surrounding  medium  and  the 
air;  then,  if  MM'  be  the  axis  of 
incidence,  a  ray  A  A'  parallel  to 
an  optic  axis  will  be  refracted  at 
A.  If  further  we  place  Z  A'  AM' 
equal  to  Fa(as  heretofore,  half  the 
axial  angle)  and  Z  MAB  equal  to 
Ha  (since  MM'  is  the  bisectrix, 
this  is  half  the  apparent  axial 
angle  in  the  surrounding  medium)  , 
letting  the  velocity  of  light  in 
,  in  the  surrounding  medium  Vh,  and  in  the  air 


the  crystal  be 
v,  we  have 


sn 


sn 


v 


From  this  equation,  since  —  is  the  intermediate  refractive  in- 
dex ft  and  -  -  the  refractive  index  from  air  into  the  enveloping 
medium,  which  we  shall  call  n,  there  follows 

I  •          T/  W  •          TT       *  /      \ 

-  •   n,    or  sin  Va  =  -  .  sin  Ha.  (i) 


sin  V, 


sin  Ha      ft  r 

According  to  this  the  true  axial  angle  may  be  calculated  even 
in  a  case  such  that  the  axes  no  longer  emerge  into  air,  if  we  sur- 
round the  crystal,  whose  intermediate  refractive  index  we  know, 
with  a  highly  refractive  medium  whose  refractive  index  n  for  the 
color  used  is  likewise  known  and  determine  the  angle  2Ha 

*  Since   sin  E  =  /?  •  Va,   then   if   in    the   above  equation  we    substitute    for 

sin  E  sin  E 

sin  Va   its  value   — —  ,  we  obtain  n  =   - — —  •      According   to   this  we  can, 
p  sin  Ha 

by  measuring  the  apparent  axial  angle  in  air  and  in  the  fluid  above-mentioned, 
determine  the  refractive  index  n  of  that  fluid  by  means  of  one  and  the  same 
crystal  plate. 


DETERMINATION  OF  OPTIC  AXES   AND   THEIR  ANGLE      191 


included  between  the  axes  in  that  medium.     This  latter  is  done 
in  the  following  way:  — 

One  surrounds  the  crystal  plate  with  a  vessel  TLH'H'H 
(Fig.  99),  whose  front  and 
whose  back  wall,  HH  and 
H'H' ',  each  consist  of  a  plane- 
parallel  glass  plate,  and  fills 
the  vessel  up  with  bromnaph- 
thalin — whose  refractive  in- 
dex is  known  —  so  that  the 
plate  is  entirely  covered,  it  be- 
ing at  the  same  time  connected 
with  the  apparatus  for  measur- 
ing the  axial  angle  in  wholly  the 
same  way  as  though  the  appar- 
ent angle  in  air  were  to  be  de- 
termined. If,  then,  one  rotates 


H' 


Fig.  99. 


the  plate  until  those  rays,  AB  (the  notation  is  entirely  the  same 
as  in  the  last  figure),  that  in  the  crystal  move  along  an  optic 
axis  are  parallel  to  the  axis  of  the  polariscope,  they  suffer  a  de- 
viation neither  at  the  boundary  of  the  oil  with  the  enclosing  glass 
plate  nor  because  of  the  plate  itself,  since  this  stands  perpendic- 
ular to  the  axis  of  the  instrument;  consequently  this  position  is  to 
be  found  in  wholly  the  same  way  as  in  measuring  the  apparent 
axial  angle  in  air  —  by  adjusting  the  dark  hyperbola  on  the  inter- 
section of  the  cross-hairs  in  the  field  of  the  polariscope.  Hence, 
if  one  rotates  back,  and  in  the  opposite  direction  until  the  second 
axial  figure  appears  centered  in  the  field  in  the  same  way,  the 
entire  rotation  necessary  for  this  is  obviously  ^Ha\  i.e.  the  ap- 
parent axial  angle  in  the  bromnaphthalin.  Therefore,  /?  and  n 
being  known,  the  true  axial  angle  follows  from  the  Ha  thus 
measured,  according  to  the  equation 


sin  Vn  =  -  •  sin  H 


a, 


derived  on  the  opposite  page. 


OPTICALLY   BIAXIAL   CRYSTALS 


-T1H 


The  same  determination,  finally,  can  be  made  even  without 
knowing  /?  and  «;  namely,  with  the  aid  of  a  second  crystal  plate, 
whose  faces  are  cut  perpendicular  to  the  bisector  of  the  obtuse 

axial  angle,  the  so-called  obtuse 
bisectrix.  From  such  a  plate 
the  axial  rays  will  in  general  no 
longer  emerge  into  air;  but  into 
bromnaphthalin  they  will,  even 
when  the  obtuse  axial  angle  is 
very  large,  provided  only  the  re- 
fractive index  of  the  liquid  is  at 
least  as  large  as  of  the  crystal. 
,  In  Fig.  100  let  there  be  repre- 
sented such  a  plate  in  the  vessel 
of  bromnaphthalin,  this  plate 
likewise  rotatable  about  the  nor- 
mal to  the  optic  axial  plane;  let 
also  A' A  be  a  ray  that  in  the  crystal  moves  parallel  to  an  optic 
axis,  being  transmitted  in  the  bromnaphthalin  along  the  direction 
AB.  Then,  if  MM'  be  the  normal  to  the  plate,  i.e.  the  obtuse 
bisectrix,  the  angle  A' AM'  is  V0,  or  half  the  true  obtuse  axial 
angle,  and  Z  MAB  is  HOJ  or  half  the  apparent  obtuse  axial 
angle  in  bromnaphthalin.  Hence  the  measurement  of  this  latter 
angle  is  made  entirely  as  with  the  last  plate  — by  rotation,  with 
successive  adjustment  of  the  two  axial  figures.  If  for  the  light 
velocity  and  the  refractive  indices  we  retain  the  same  notation  as 
above  with  the  acute  axial  angle,  then  here,  in  wholly  the  same 
way  as  there,  we  obtain 


Fig.  100. 


sin  V0 
sin  Hf 


sin  Vn 


n       .     ,T 
-  •  sin  H, 


(2) 


By  means  of  this  equation,  therefore,  when  the  apparent  obtuse 
axial  angle  in  bromnaphthalin  has  been  determined,  the  true  may 


DETERMINATION   OF  OPTIC  AXES  AND  THEIR  ANGLE      193 

be  calculated  from  it,  just  as,  by  means  of  the  previously  de- 
veloped equation,  from  the  apparent  acute.  But  both  calcula- 
tions presuppose  a  knowledge  of  the  intermediate  refractive 
index  of  the  crystal  and  of  the  refractive  index  of  the  oil, 
Since,  however,  the  sum  of  the  acute  and  the  obtuse  axial  angle 
for  the  same  color  must  always  be  180°,  then  Va  +  V0  =  90° 
and  therefore  sin  V0=  cos  Va.  If  we  substitute  this  value  in 
equation  (2)  and  divide  the  resulting  equation  into  equation  d) , 
developed  on  page  190  for  the  acute  axial  angle, 

_.  \ 

sin  Va  =  -  •  sin  Ha\  (i) 

f  cos  Va  =  -  •  sin  HA  (2) 

we  obtain  .     rr 

tang  V,  =  %f?  • 
sm  Ho 

That  is,  we  can  determine  the  true  optic  axial  angle  of  a  crystal 
without  knowing  any  refractive  index;  for,  if  we  cut  from  it  two 
plates,  one  of  them  perpendicular  to  the  acute,  the  other  to  the 
obtuse  bisectrix,  and  determine  with  both  plates  in  the  manner 
described  the  apparent  axial  angle  in  bromnaphthalin,  the  quo- 
tient of  the  sines  of  these  angles  is  the  tangent  of  half  the  interior 
axial  angle  sought.  This  method  of  determining  the  angle  is  of 
special  importance  because,  for  the  preparation  of  prisms  with 
which  the  refractive  indices  can  be  very  accurately  measured, 
transparent  crystals  are  required  of  a  size  that,  by  far,  is  not  to 
be  had  with  all  substances;  while  the  plane-parallel  plates  for 
this  method  may  be  of  almost  any  smallness*  and,  moreover,  can 
be  more  easily  prepared  in  sufficient  exactness  than  can  cor- 
rectly oriented  prisms.  When,  therefore,  only  very  small  crys- 
tals may  be  had  for  determination  of  the  refractive  indices,  one 
must  be  content  with  measuring  the  true  axial  angle,  by  the 

*  Even  for  determining  the  refractive  indices  with  the  total-reflectometer  the 
crystals  must  not  be  too  very  small,  because  otherwise  the  light  reflected  from  the 
plate  is  so  faint  that  the  boundaries  of  total  reflection  can  no  longer  ba  discerned. 


IQ4  OPTICALLY   BIAXIAL   CRYSTALS 

method  described;  and,  as  for  the  rest,  the  intermediate  refrac- 
tive index  /?  is  obtained  if  by  means  of  the  plate  perpendicular 
to  the  acute  bisectrix  one  determines  the  apparent  axial  angle 
2E  in  air,  according  to  the  equation  (see  p.  188) 

r>  n          •        -IT  n  sin   R 

sin  E  =  8  •  sin  Va,    or    8  =  - — -• . 

sin  Va 

But  if  /?  is  known,  then,  by  measuring  (with  the  aid  of  the  polari- 
zation-colors) the  birefringence  of  two  plates  of  any  smallness 
that  are  parallel  to  XY  and  YZ  respectively,  one  can  determine 
in  addition  the  differences  7-— /?  and  j)  —  a,  and  thereby  f  and  a.* 
Instead  of  a  liquid  of  strong  refraction,  for  measuring  the 
acute  and  the  obtuse  apparent  axial  angle,  we  may  use  a  highly 
refractive  solid  body.  On  this  principle  is  based  the  measure- 
ment of  axial  angles  by  means  of  W.  G.  Adams's  apparatus.! 
With  this  method  the  plane-parallel  crystal  plate  lies  between 
the  plane  faces,  which  stand  vis-a-vis,  of  two  lenses  of  highly 
refractive  glass;  the  other  two  lens  faces,  after  the  lenses  are 
put  together,  form  a  sphere;  and  this  sphere  is  rotatable  sepa- 
rately from  the  rest  of  the  apparatus.  Rays  that  have  passed 

*  With  the  aid  of  the  quantities  V  and  /?  we  can  furthermore  determine  the 
complete  optical  constants  of  a  crystal  by  measuring  the  interval  between  the 
lemniscates  that  are  exhibited  by  a  plate  of  known  thickness  cut  perpendicular 
to  the  acute  bisectrix.  For  when  this  distance  is  known  it  is  possible  (see  A. 
Muttrich,  Pogg.  Ann.  d.  Physik,  1864,  121,  206  et  seq.)  to  calculate  the  value 

— 2  —  -5 ;  so  that,  if  this  value  is  substituted  in  the  equation 


f 

which  holds  good  for  the  optic  axial  angle  just  as  does  that  on  p.  143,  there  remains 
only  one  unknown  quantity,  f\  and  after  the  latter  has  been  calculated,  a  follows 
directly  from  the  same  formula. 

f  Described  and  illustrated  by  figures  in  Phys.  Kryst.  4th  ed.  753-759;  3rd 
ed.  723-729.  See  also  the  following  literature:  W.  G.  Adams,  Proceed.  Phys.  Soc. 
1,  152;  Phil.  Mag.  1875,  50,  and  1879,  [5]  8,  275;  and  in  Zeitschr.  f.  Kryst.  5, 
381;  further,  E.  Schneider,  Carls  Repert.  /.  Exper.-Physik,  15,  774  (1879);  F.  Becke, 
Tscher.  min.  u.  petrogr.  Mitteil.  1879,  2,  430.  And  see  in  addition  Leiss:  Die  op. 
Instrum.,  166  et  seq. 


DETERMINATION  OF  OPTIC  AXES  AND   THEIR   ANGLE      195 

through  the  crystal  plate  lying  at  the  center  of  the  sphere  ex- 
perience, on  emerging  from  the  latter,  no  further  deviation;  and 
consequently,  by  adjusting  the  two  axial  figures  a  measure- 
ment is  made  of  the  acute  or  of  the  obtuse  apparent  axial  angle 
in  the  kind  of  glass  in  question  (instead  of  in  bromnaphthalin). 

When  the  optic  axial  angles  of  a  biaxial  crystal  are  deter- 
mined for  different  colors,  whether  by  complete  measurement 
of  the  optical  constants  (of  the  three  principal  refractive 
indices)  or  by  their  direct  determination,  they  are  found  to 
differ;  and  moreover,  the  size  of  the  axial  angle  rises  or  falls 
steadily  with  the  wave  length  of  the  light  to  which  the  axes 
refer.  Since  each  of  the  three  principal  refractive  indices  varies 
with  the  color  approximately  after  the  same  law,  stated  on  page 
46  as  Cauchy's  dispersion  formula  for  singly  refracting  media, 
except  that  naturally  with  each  index  the  constants  of  this  formula 
have  a  different  value,  the  natural  conjecture  is  that  the  axial 
angles,  too,  vary  with  the  color  according  to  a  similar  law.  As  a 
matter  of  fact,  the  axial  angles  of  those  crystals  in  which  the 
angle  increases  with  the  wave  length  (sense  of  the  dispersion 
p  >  u)  correspond  with  remarkable  approximation  to  the  formula 

v.-A-S, 

and  the  angles  of  those  whose  axial  angle  diminishes  with  greater 
wave  length  of  the  light  (p  <  u},  to  the  formula 


When,  therefore,  one  has  determined  the  true  axial  angle  2  Va  of 
a  substance  for  two  colors  of  known  wave  length,  then  by  sub- 
stituting the  values  found,  in  the  proper  one  of  these  two  equa- 
tions, one  can  deduce  the  constants  A  and  B  for  the  body,  and 
from  them  the  axial  angle  for  every  other  wave  length.  It  must 
be  remarked,  however,  that  some  few  substances  (e.g.  gypsum) 
present  exceptions  to  this  rule,  having  an  abnormal  dispersion  of 
the  optic  axes,  such  that  their  angle  exhibits  a  maximum  or  a 
minimum  for  a  definite  intermediate  color. 


196  RECAPITULATION 


RECAPITULATION:    CLASSIFICATION    OF     CRYS- 
TALS ACCORDING  TO  THEIR  OPTICAL 
PROPERTIES 

As  follows  from  what  has  preceded,  the  relation  in  which 
the  transmission  of  light  in  a  crystal  stands  to  the  crystal- 
lographic  direction  can  in  general  be  represented  by  a  triaxial 
ellipsoid,  wherefore  the  optical  properties  of  crystals  belong 
with  the  ellipsoidal  properties.  (See  p.  7.)  The  most  general 
case  is  that  treated  on  page  181  et  seq.,  in  which  the  index- 
surface  has  for  the  different  colors  not  only  a  different  form, 
but  also  a  different  crystallographic  orientation  of  its  three 
principal  axes.  A  special  case  is  presented  by  those  crystals 
(see  p.  174  et  seq.)  whose  index-surfaces  for  the  different  colors 
have  one  axis  in  common;  an  additional  one  by  those  in  which 
all  three  principal  axes  of  all  the  ellipsoids  corresponding  to  the 
different  colors  are  oriented  alike  in  the  crystal.  Still  more 
special  is  the  case  in  which  two  principal  axes  of  the  index- 
surfaces  become  of  equal  length,  the  surfaces  passing  over  into 
ellipsoids  of  rotation.  And  finally  the  last,  most  special  case 
lies  before  us  when  all  three  principal  axes  of  these  surfaces  be- 
come equal;  i.e.  when  the  index-surfaces  for  all  colors  assume 
the  same  (spherical)  form. 

Thus,  for  the  totality  of  crystals,  there  results  a  division 
according  to  their  optical  properties  into  the  following  five 
groups:— 

1.  Biaxial  crystals  with  no  plane  of  optical  symmetry, — in 
which  crystals  no  two  straight  lines  of  different  orientation  are 
optically  equivalent,  two  different  directions  being  equivalent 
only  when  they  are  opposite,  lying  therefore  in  the  same  straight 
line.     (That  two  such  directions  are  equivalent  is  because  the 
optical  properties  belong  with  the  bi-vector  —  see  p.  14.) 

2.  Biaxial  crystals  having  one  plane  of  optical  symmetry.     In 
these  crystals  there  exists  for  every  direction  a  direction  that 


RECAPITULATION  197 

in  optical  respects  is  equivalent  to  it,  .the  latter  direction  lying 
symmetrical  with  the  former,  with  reference  to  that  plane  of 
symmetry;  here,  therefore,  counting  the  two  directions  opposite 
to  those  mentioned,  there  are  always  four  optically  equivalent 
directions,  and  these  lie  in  two  straight  lines  whose  angle  is 
bisected  by  the  plane  of  symmetry. 

3.  Biaxial  crystals  having  three  mutually  perpendicular  planes 
of  optical  symmetry.     To  a  direction  of  any  orientation  there 
here  belong,  as  equivalent,  yet  seven  others;  these  lie  in  four 
straight  lines,  which  always  have,  pairwise,  equal  and  opposite 
inclination  to  the  three  planes  of  symmetry. 

4.  Uniaxial  crystals.     These  have  an  infinity  of  planes  of 
optical  symmetry,  which  all  intersect  in  the  axis.     Crystals  of 
this  kind  exhibit  the  same  behavior  in  all  directions  (infinite  in 
number)  that  include  the  same  angle  with  the  axis. 

5.  Singly  refracting  crystals.     Here  all  directions  (an  infinity 
therefore  in  a  higher  sense  than  in  the  last  case)  are  optically 
equivalent.* 

Each  of  the  first  four  groups  embraces  crystals  with  positive 
double  refraction  and  crystals  with  negative  double  refraction; 
yet  this  difference  among  crystals  is  not  essential  theoretically, 
since  for  different  colors  the  same  crystal  can  have  double 
refraction  of  opposite  character,  nor  is  the  distinction  requisite 
for  the  systematic  classification.  Practically,  however,  for  the 
determination  of  crystallized  bodies  by  their  optical  properties, 
it  is  important  to  tell  the  positively  and  the  negatively  doubly 
refracting  apart.  The  methods  adapted  for  this  purpose  are 
elucidated  in  the  following  section. 

*  [Group  i  comprises  the  triclinic  crystal  system;  2,  the  monoclinic;  3,  the  ortho- 
rhombic;  4,  the  hexagonal,  the  trigonal  (Dana's  trigonal  division  of  the  hexagonal 
system),  and  the  tetragonal;  5,  the  cubic,  or  isometric.] 


I  g8      COMBINATIONS    OF  DOUBLY   REFRACTING   CRYSTALS 

COMBINATIONS  OF  DOUBLY  REFRACTING 
CRYSTALS 

DETERMINATION  OF  THE  CHARACTER  OF  THE  DOUBLE  RE- 
FRACTION OF  UNIAXIAL  AND  BIAXIAL  CRYSTALS  BY 
COMBINATION  WITH  OTHER  DOUBLY 
REFRACTING  CRYSTALS 

As  was  shown  on  page  63  et  seq.,  it  is  possible  to  determine 
the  birefringence  of  a  thin  crystal  plate,  i.e.  the  difference  of 
path  of  the  two  light  rays  transmitted  in  such  a  plate,  by  means 
of  the  arising  interference-color.  This  determination  is,  to  be 
sure,  only  approximate,  since  the  color  is  further  influenced  by 
the  diversity  of  the  double  refraction  for  different  colors;  there 
is  found  only  a  mean  value,  as  it  were,  for  the  retardation  that 
has  taken  place  in  the  crystal.  From  this  retardation  and  the 
plate  thickness  we  can  find  the  difference  between  the  refractive 
indices  for  the  two  interfering  rays  by  the  formula  given  in  the 
footnote  on  page  63.  But  herewith  it  remains  undetermined 
which  of  the  two  rays  has  the  greater,  which  the  lesser  velocity; 
in  other  words,  what  is  the  character  of  the  double  refraction 
of  the  crystal  —  whether  positive  or  negative.  To  determine 
this  it  is  necessary  to  combine  the  plate  with  a  second  crystal 
plate,  the  positive  or  negative  character  of  whose  double  refrac- 
tion is  known.  When  it  is  a  matter  of  very  thin  crystal  sections, 
such  as  usually  form  the  subject  of  microscopical  investigations 
and  which,  being  so  thin,  exhibit  colors  only  of  the  lower  orders, 
the  most  suitable  plate  for  determining  the  sign  of  the  double 
I/  refraction  is  a  thin  plate  of  gypsum,  a  plate  that  between  crossed 
nicols  exhibits  the  red  of  the  first  order  (see  p.  66) ;  in  cases 
>/of  very  low  birefringence  we  may  use  instead  a  so-called  quarter- 
undulation  mica  plate,  —  i.e.  a  cleavage-lamella  of  mica  so  thin 
that  the  two  vibrations  arising  from  a  ray  entering  it  at  right 
angles  are  retarded,  the  one  as  compared  with  the  other,  only 
%L*  Mica  is  negatively  biaxial,  that  is  to  say,  the  vibration 

*  Strictly  speaking,  this  holds  true  only  for  one  definite  color;  but,  for  the 


DETERMINATION  OF  OPTICAL   CHARACTER  199 

direction  of  its  greatest  light  velocity  bisects  the  acute  angle  of 
the  axes;  and  the  so  extremely  perfect  cleavage  of  mica  is  almost 
exactly  perpendicular  to  that  vibration  direction.  If  the  sheet 
of  mica  is  cut  into  a  rectangular  form  so  that  the  long  direc- 
tion corresponds  to  the  optic  axial  plane,  its  longer  edges  are 
then  parallel  to  the  vibration  direction  of  the  least  light  velocity, 
the  shorter  to  that  of  the  intermediate;  and  these  directions  are 
at  the  same  time  the  two  vibration  directions  of  rays  emerg- 
ing perpendicular  to  the  plate  (hence  parallel  to  the  acute 
bisectrix). 

When  such  a  mica  plate  is  so  introduced  into  the  path  of  the 
light  rays,  before  they  enter  the  analyzer,  that  its  long  direction 
bisects  the  angle  formed  by  the  vibration  directions  of  the  two 
crossed  nicols  (for  which  purpose  the  microscope  must  be  fitted 
with  a  suitable  slit),  the  field  is  lighted  up  with  a  blue-gray 
color,  corresponding  to  the  JA  for  medium  colors, — to  a  path 
difference,  therefore,  amounting  to  about  140  ///*.  (Cf.  p.  64.) 
Now  supposing  there  is  present  in  the  field  of  view  a  very  thin 
crystal  of  extremely  low  birefringence,  in  the  position  where  its 
vibration  directions  coincide  with  those  of  the  nicols,  it  can  pro- 
duce no  change  and  therefore  appears  lighted  up  just  like  the 
rest  of  the  field.  But  if  one  rotates  the  crystal  45°  in  its  own 
plane,  there  takes  place  in  it  a  resolution  of  the  light  into  two 
vibrations  of  very  small  difference  of  path.  If  the  sense  of  the 
rotation  was  such  that  the  direction  of  the  vibration  that  in  the 
crystal  is  transmitted  with  the  greater  velocity  coincides  with 
the  vibration  direction  of  the  ray  that  has  the  greater  velocity  in 
the  mica  plate  (in  which  case  therefore  the  two  vibration  direc- 
tions of  the  rays  transmitted  the  slower  in  the  two  crystals,  re- 
spectively, also  become  parallel) ,  then  to  the  path  difference  arising 
in  the  crystal  there  will  be  added  that  (J^)  arising  in  the  mica;  con- 
method?,  treated  in  the  following,  it  is  sufficient  if  the  retardation  amounts  to  |A 
of  a  medium  color,  since  for  the  remaining  colors  it  is  then  but  little  different  from 
$L  "Achromatic"  retardation  plates,  which  yield  exactly  JA  difference  of  path 
for  all  colors,  can  be  produced  by  combining  different  plates.  (Se2  Zeitschr.  f. 
Kryst.  1903,  37,  292.) 


200      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

sequently  the  blue-gray  color  passes  over  into  a  brighter,  greenish 
gray,  or,  when  the  crystal  is  somewhat  more  birefringent,  as  far  as 
into  yellowish  white.  Even  when  the  double  refraction  of  the  crys- 
tal is  extremely  slight,  it  can  be  perceived  by  this  method,  owing 
to  the  greater  brightness  on  rotation.  If,  however,  one  rotates 
the  crystal  45°  in  the  opposite  sense,  then  does  the  vibration 
direction  of  the  ray  transmitted  the  faster  in  the  crystal  coincide 
with  the  direction  of  the  vibration  that  in  the  mica  advances  the 
slower,  and  vice  versa.  The  total  difference  of  path  becomes, 
in  consequence,  just  as  much  smaller  as  it  previously  became 
larger;  so  the  order  of  the  interference-color  diminishes;  that 
is,  the  crystal  becomes  darker,  and  lavender-gray.  But,  through 
observation  of  the  crystal  to  be  investigated,  in  both  positions, 
the  problem  before  us  is  solved;  for  in  the  crystal  THAT  VIBRA- 
TION IS  ALWAYS  TRANSMITTED  THE  SLOWER  THAT  IS  PARALLEL 
TO  THE  OPTIC  AXIAL  PLANE  OF  THE  MICA  WHEN  THE  CRYSTAL 

APPEARS  BRIGHTER  THAN  THE  FIELD.  By  this  procedure,  there- 
fore, one  is  able  not  only  to  recognize  double  refraction  in  a 
crystal,  even  when  it  is  so  slight  that  without  addition  of  the 
mica  plate  the  brightening  on  rotation  between  crossed  nicols 
would  escape  observation,  but  also  to  determine  the  character 
of  the  double  refraction. 

The  same  is  the  case  when  the  crystal  is  combined  with  a 
gypsum  plate  exhibiting  with  crossed  nicols  the  red  of  the  first 
order.  If  we  insert  this  plate  into  the  orthoscope  in  wholly  the 
same  way  as  we  inserted  the  mica,  the  whole  field  appears  in  the 
color  named;  and  so  likewise  do  any  doubly  refracting  crystals 
present  in  it,  provided  their  vibration  directions  coincide  with 
those  of  the  nicols.  But  if  we  rotate  the  stage  of  the  instrument 
until  the  vibration  directions  of  a  crystal  to  be  investigated 
include  45°  with  those  of  the  nicols,  then  in  the  crystal  there 
arises  a  difference  of  path,  which  is  added  to  or  subtracted  from 
that  effected  in  the  gypsum  plate,  according  as  the  two  vibra- 
tion directions  of  the  greater  of  the  two  light  velocities  in  crystal 
and  gypsum  plate  respectively  are  parallel  or  crossed.  Conse- 


DETERMINATION  OF  OPTICAL  CHARACTER       2OI 

quently  the  crystal,  as  compared  with  the  rest  of  the  field,  ex- 
hibits a  color  of  higher  or  of  lower  order,  according  as  it  has 
been  rotated  45°  in  the  one  or  in  the  opposite  sense.  When, 
therefore,  by  comparison  with  a  crystal  of  known  double  refrac- 
tion, one  has  ascertained  for  the  rectangular  gypsum  plate 
whether  the  vibration  parallel  to  its  long  direction  is  that  trans- 
mitted the  faster  or  the  slower,  it  is  obvious  that  through  this 
change  of  color  one  can  determine  the  unknown  character  of  the 
double  refraction  of  every  crystal  present  in  the  field  of  the 
orthoscope,  —unless  the  path  difference  produced  by  the  crystal 
is  so  considerable  that  doubt  can  arise  as  to  the  order  of  the 
resulting  color  shade.  Supposing  this  to  be  the  case,  the  crystal 
when  in  diagonal  position  producing,  of  itself  alone,  a  color  of 
the  third  or  of  the  fourth  order,  then  by  inserting  the  gypsum 
plate  parallel  to  the  one  diagonal  this  color  is  transformed  into 
a  color  whose  order  is  higher  by  the  path  difference  ^,  i.e.  by 
one  whole  order;  while  if  we  insert  the  gypsum  plate  along 
the  other  diagonal,  or  rotate  the  crystal  90°,  the  order  of  the 
color  is  just  so  much  lowered.  Since  interference-colors  differ- 
ing by  two  whole  orders  can  always  be  clearly  distinguished, 
owing  to  the  circumstance  that  the  higher  is  less  vivid  than 
the  lower  and  more  nearly  approximate  to  white,  then  with 
the  aid  of  the  sensitive  gypsum  plate  the  character  of  the 
double  refraction  can,  in  such  cases  also,  be  established  beyond 
question. 

Finally,  if  in  consequence  of  considerable  thickness  or  very 
high  birefringence  a  crystal  plate  exhibits  the  white  of  a  higher 
order,  then,  to  produce  a  color  that  differs  perceptibly  from  this 
white,  the  path  difference  must  be  diminished  more  than  one 
wave  length.  For  this  purpose  it  is  best  to  use  a  doubly  refract- 
ing crystal  ground  to  the  form  of  a  wedge;  e.g.  a  quartz  wedge 
one  of  whose  faces  is  parallel  to  the  optic  axis,  A  A.  (See  Fig.  101, 
p.  202:  a,  front  view;  6,  longitudinal  section.)  The  wedge,  in 
order  to  render  the  thinnest  part  less  liable  to  break,  is  commonly 
cemented  on  a  rectangular  glass  plate,  g.  Since  the  double 


202      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 


refraction  of  quartz  is  positive,  then  in  such  a  wedge  a  ray 
falling  perpendicularly  at*  any  point  is  split  up  into  two  rays 
of  which  the  ordinary,  vibrating  parallel  to  BB,  is  transmitted 

the  faster;  the  extraordinary, 

j vibrating  parallel  to  the  axis 

A  A,  the  slower.  Now  the 
difference  of  path  with  which 
the  two  rays  emerge  from  the 
quartz  obviously  increases 
with  the  thickness  of  the  lat- 
ter; consequently,  by  shifting 
the  wedge  from  right  to 

left  we  can  increase  this  difference  of  path,  since  a  thicker  part 
of  the  wedge  then  comes  into  play.  Hence,  if  one  places  a 


doubly  refracting  crystal  plate,  dba'V  (Fig.  102),  in  the  ortho- 
scope  so  that  its  vibration  directions  form  45°  with  those  of  the 
two  crossed  nicols  (NN  and  N'N'),  and  if,  of  all  the  vibra- 
tion directions  lying  in  the  plane  aba'b',  aaf  be  that  of  the  great- 
est light  velocity  and  bb'  that  of  the  least,  then  in  the  crystal  the 


DETERMINATION  OF  OPTICAL  CHARACTER       203 

vibrations  parallel  to  aa'  will  be  transmitted  faster  than  those 
parallel  to  W\  accordingly  the  two  rays,  when  they  emerge, 
will  have  a  difference  of  path;  — and  let  this  difference,  n\,  be 
of  a  magnitude  such  that  the  white  of  a  higher  order  results. 
If,  then,  the  quartz  wedge,  QQQ'Q' ',  is  so  inserted  that  its  optic 
axis,  A  A,  is  parallel  to  the  vibration  direction  aa'  of  the  crystal 
plate,  each  of  the  two  rays  emerging  from  the  crystal  will  be 
transmitted  in  the  quartz  with  unaltered  vibration  direction; 
but  the  vibration  parallel  to  aa'  here  advances  the  slower,  the 
vibration  parallel  to  bb'  the  faster;  consequently  the  path  differ- 
ence acquired  in  the  quartz  —  let  it  be  n'X  —  is  of  the  opposite 
sense  to  that  arising  in  the  crystal,  and  after  the  two  rays  have 
passed  through  both  crystal  and  quartz  their  path  difference  is 
thus  (n  —  n')L  The  quantity  n  —  ri  can  be  made  as  small  as 
desired,  provided  the  crystal  plate  is  rather  thin  and  the  quartz 
wedge  thick  enough,  by  shifting  the  latter  parallel  to  A  A,  so  that 
a  thicker  part  comes  to  the  center.  With  this  diminished  path 
difference,  then,  the  two  rays  enter  the  analyzer,  and  after  the 
reduction  there  to  one  vibration  plane  they  interfere  in  a  manner 
corresponding  to  this  path  difference.  When  n  —  n'  is  very 
small,  wholly  the  same  conies  to  pass  as  though  the  crystal  plate 
were  even  extremely  thin  and  no  quartz  wedge  present;  that  is, 
vivid  interference-colors  appear. 

On  the  other  hand,  should  the  quartz  wedge  have  been  so 
inserted  in  the  polariscope  that  A  A  were  parallel  to  the  vibra- 
tion direction  bb',  the  same  vibrations  that,  as  compared  with 
those  perpendicular  to  them,  were  in  the  crystal  retarded  n\ 
would  then,  in  the  quartz  likewise,  experience  a  relative  retarda- 
tion, of  w'A;  in  the  end,  therefore,  they  would  have  a  path  differ- 
ence of  (n  +  n'}X  and  interfere  correspondingly.  So  in  this 
case  the  quartz  wedge  operates  as  though  the  crystal  plate  had 
become  thicker,  and  consequently  a  more  nearly  perfect  white 
of  a  higher  order  appears. 

If  the  vibration  direction  of  the  greatest  light  velocity  in  the 
crystal  plate  were  not  aa',  as  we  have  assumed,  but  bb',  and  aaf 


204      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

that  of  the  least,  then  everything  would  be  reversed;  that  is, 
to  obtain  the  interference-colors  we  should  have  to  so  insert 
the  quartz  wedge  that  A  A  were  parallel  to  bb'. 

This  procedure  for  determining  the  character  of  the  double 
refraction  may  be  employed  also  in  the  investigation  of  a  crystal 
in  convergent  light.  Here,  unless  the  interference-figure  of  an 
optic  axis  happens  to  appear  in  the  field,  a  plate  of  some  thick- 
ness exhibits  no  bright  and  dark  curves  as  in  monochromatic 
light,  but  merely  the  white  of  a  higher  order.  If  by  rotating  the 
stage  of  the  conoscope  the  crystal  plate  is  turned  so  that  it 
exhibits  the  maximum  brightness  (its  vibration  directions  then 
form  45°  with  those  of  the  crossed  nicols),  and  if  between  crys- 
tal and  analyzer  the  quartz  wedge  is  now  inserted  once  with  its 
long  direction  parallel  to  the  one  vibration  direction  of  the 
crystal  plate,  and  afterward  parallel  to  the  other,  — then,  of 
these  two  vibration  directions,  the  one  to  which  the  long  direc- 
tion of  the  wedge  is  parallel  when  hyperbolic  color  curves  appear 
in  the  center  of  the  field  is,  of  all  vibration  directions  parallel 
to  the  plate,  that  of  the  greatest  light  velocity,  the  vibration 
direction  standing  perpendicular  to  it  that  of  the  least  light 
velocity* 

If  we  have  at  our  disposal,  for  the  determination  of  its  op- 
tical character,  a  uniaxial  crystal  plate  whose  faces  stand  per- 
pendicular to  the  optic  axis,  the  determination  can  be  made  by 
laying  the  plate  on  the  stage  of  the  conoscope,  and  upon  this 
plate  a  second,  cut  perpendicular  to  the  axis  from  another 
uniaxial  crystal,  the  sign  of  whose  double  refraction  is  known. 
When  the  plate  to  be  investigated  has  the  same  optical  char- 
acter as  the  latter,  the  same  ray  (of  the  two  vibrating  perpen- 
dicularly to  each  other)  that  was  retarded  in  the  lower  plate 
is  retarded  in  the  upper  one  as  well;  so  this  upper  plate  operates 
exactly  as  though  the  lower  had  become  thicker.  In  other 
words,  the  circular  color  rings  will  become  narrower  than  they 
appeared  before  the  plate  of  known  optical  character  was  su- 
perposed. When  on  the  other  hand  the  crystal  plate  to  be  in- 


DETERMINATION  OF  OPTICAL   CHARACTER 


205 


vestigated  is  of  opposite  character  to  the  known,  the  latter  plate 
will  operate  as  though  the  former  had  become  thinner;  that  is, 
the  color  rings  will  become  wider.  This  widening  or  narrow- 
ing of  the  color  rings  can  very  easily  be  made  perceptible,  by 
placing  in  the  focal  plane  of  the  polariscope  a  glass  plate  hav- 
ing fine  lines  scratched  upon  it;  these  lines  are  then  seen  on  the 
interference-figure,  and  we  thus  have  a  measure  for  determining 
the  diameter  of  the  color  rings. 


II 


's. 


IV 


/  /M  > 
/ 


-N' 


in 


N 

Fig.  103. 


A  further  very  convenient  method  —  and,  in  fact,  the  one 
most  frequently  employed  —  of  determining  the  optical  char- 
acter of  a  uniaxial  plate  cut  perpendicular  to  the  axis,  consists 
in  the  use  of  the  quarter -undulation  mica  plate,  described  on  page 
198.  If  this  is  introduced  into  the  path  of  the  light  rays,  between 
crystal  plate  and  analyzer,  then,  in  homogeneous  light,  instead 
of  the  circular  dark  rings  with  the  black  cross  we  obtain  when 
the  crystal  is  positive  the  interference-figure  shown  in  Fig.  103, 
but  when  the  crystal  is  negative  the  one  shown  in  Fig.  104  (p. 
206), —  provided  that  in  both  cases  the  mica  sheet  has  the  diag- 
onal position  indicated  by  dots;  i.e.  that  its  long  axis  forms  45° 
with  the  nicols  in  the  two  quadrants  marked  //  and  IV.  The 


206      COMBINATIONS    OF    DOUBLY   REFRACTING   CRYSTALS 


former  interference-figure  differs  from  the  usual  one  in  these 
particulars:  the  dark  rings  in  quadrants  77  and  IV  are  narrowed 
to  the  extent  of  about  quarter  the  interval  between  two  adja- 
cent rings,  while  those  in  quadrants  7  and  777  are  just  so  much 
widened,  the  result  being  that  at  the  boundary  of  two  quad- 
rants a  bright  ring  always  abuts  against  a  dark;  instead  of  the 
black  cross,  two  black  spots  appear,  and  the  line  connecting 
these  spots  stands  perpendicular  to  the  long  direction  of  the 
m?ca.  Should  the  mica  sheet  have  been  inserted  perpendicular 


II 


IV 


III 


to  the  position  indicated  in  the  figure,  so  that  its  long  direction 
had  fallen  in  the  middle  of  quadrants  7  and  777,  then  would 
the  rings  of  these  quadrants  be  narrowed,  those  of  77  and  IV 
widened,  and  in  the  latter  quadrants  the  black  spots  would 
appear.  In  the  case  of  Fig.  104,  representing  the  interference- 
figure  of  a  negative  crystal,  the  same  widening  of  the  rings, 
together  with  the  dark  spots,  lies  in  those  quadrants  that  are 
bisected  by  the  long  axis  of  the  mica,  i.e.  in  77  and  IV,  but  the 
narrowing  of  the  rings  occurs  in  7  and  777.  Should  the  mica 
sheet  have  been  so  inserted  that  it  bisected  quadrants  7  and  777, 
then  the  widening  of  the  rings,  together  with  the  dark  spots, 


DETERMINATION  OF  OPTICAL   CHARACTER  207 

would  appear  in  these  quadrants,  the  narrowing  in  II  and  IV. 
What  takes  place  for  the  dark  and  bright  rings  in  homogeneous 
light  holds  true  also  of  the  color  rings  appearing  in  white  light, 
so  that  accordingly  in  the  latter  case  the  isochromatic  rings  are 
widened  or  narrowed  in  the  same  way;  the  black  spots  appear 
just  the  same  as  in  homogeneous  light.  It  is  sufficient,  there- 
fore, to  explain  the  phenomenon  for  monochromatic  light. 

As  before,  let  NN,  N'N'  (Fig.  103)  be  the  vibration  direc- 
tions of  the  nicols,  MM'  the  long  direction  of  the  mica  sheet; 
then  may  the  dotted  circle  represent  the  locus,  in  the  field,  of 
the  first  dark  interference-ring  that  would  appear  if  no  quarter- 
undulation  plate  were  present.  When  the  crystal  to  be  inves- 
tigated is  optically  uniaxial,  this  ring,  according  to  a  previous 
page,  arises  in  consequence  of  two  rays  interfering  with  each 
other  in  the  corresponding  direction;  one  of  these  rays  (the 
extraordinary)  vibrates  in  the  principal  section,  while  the  other 
(the  ordinary),  vibrating  perpendicularly  to  the  former,  is 
transmitted  faster  than  it  by  an  amount  such  that  when  the  two 
rays  emerge  they  have  a  path  difference  of  one  whole  wave 
length.  For  then,  since  with  crossed  nicols  the  interference 
takes  place  with  opposite  state  of  vibration,  total  extinction 
occurs.  At  a  point  a,  where,  somewhat  nearer  the  center  of 
the  field,  the  black  quarter  circle  is  given,  the  one  ray  is  retarded 
only  |  A  as  compared  with  the  other.  But  if  we  suppose  the 
two  rays  to  enter  the  mica  sheet,  this  retardation  is  increased 
an  additional  \X\  for  the  extraordinary  ray,  vibrating  in  the 
principal  section  of  the  crystal  and  therefore  parallel  to  the  axial 
plane  MM'  of  the  mica,  is  in  this  latter,  likewise,  transmitted 
slower  than  the  ray  vibrating  perpendicularly  to  it,  since  the 
direction  MM'  is  the  vibration  direction  of  least  light  velocity 
in  the  mica,  and  SS'  that  of  the  intermediate.  Thus,  after 
addition  of  the  quarter-undulation  plate  the  two  interfering  rays 
have  the  path  difference  of  one  whole  wave  length  at  a  smaller 
distance  from  the  center  of  the  field;  so  the  first  dark  ring, 
which  arises  with  this  retardation,  has  become  correspond- 


208      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

ingly  narrower.  Let  us  now  consider  the  point  6,  at  which  rays 
converge  whose  difference  of  path  without  the  mica  plate  would 
be  |^.  Here  the  extraordinary,  the  more  slowly  transmitted,  ray 
vibrates  parallel  to  SS'  (in  the  principal  section) ;  the  ordinary, 
the  more  rapidly  transmitted,  parallel  to  MM'.  But  in  the  mica 
the  velocity  of  the  latter  ray  is  the  lesser,  that  of  the  former 
the  greater.  Accordingly  the  path  difference,  |  X,  that  arose  in 
the  crystal  plate  is  diminished  \X,  in  consequence  whereof  the 
first  dark  ring  in  quadrant  /  does  not  appear  until,  in  passing 
out  from  the  center,  we  reach  the  distance  of  the  point  b:  it  has 
become  wider  than  before.  In  wholly  the  same  way,  for  every 
succeeding  ring  in  quadrants  //  and  IV  there  is  joined  to  the 
path  difference  of  the  crystal  plate  an  additional  J^  on  the  part 
of  the  mica,  and  these  rings  all  become  narrower;  while  with 
all  the  rings  in  quadrants  7  and  ///  the  path  difference  arising 
in  the  plate  is  by  the  mica  diminished  \X,  so  that  the  latter  rings 
become  wider.  In  the  case  of  negative  crystals  the  transmis- 
sion velocity  of  the  rays  vibrating  parallel  to  the  axis  is  the 
greatest,  and  the  extraordinary  ray  therefore  the  faster;  con- 
sequently, with  the  mica  in  the  same  position,  exactly  the  same 
must  occur  in  quadrants  II  and  IV  as  took  place  with  the  posi- 
tive crystals  in  quadrants  /  and  ///,  and  vice  versa. 

In  those  quadrants  in  which  the  path  difference  acquired  in 
the  crystal  is  by  the  mica  diminished  \X  (in  which  therefore  the 
rings  are  widened),  the  darkness  corresponding  to  the  path 
difference  zero  must  arise  at  that  distance  from  the  center  at 
which  the  path  difference  without  insertion  of  the  mica  plate 
would  be  JA;  consequently,  in  these  quadrants  dark  spots  must 
appear  at  the  distance  mentioned.  The  center  of  the  figure, 
and  likewise  the  points  at  which  without  insertion  of  the  mica 
the  arms  of  the  black  cross  would  appear,  must  be  bright;  be- 
cause there  the  plate  is  traversed  by  only  one  vibration,  and  this 
is  resolved  in  the  mica  into  two  equal  components  with  J  A  differ- 
ence of  path.  These  latter  then  combine  to  form  a  circular  vi- 
bration (see  p.  20),  which  cannot  be  extinguished  by  the  analyzer. 


DETERMINATION  OF  OPTICAL   CHARACTER 


209 


The  practical  procedure  for  determining  the  character  of  the 
double  refraction  of  a  uniaxial  crystal  plate  cut  perpendicular 
to  the  axis  consists,  accordingly,  in  inserting  between  it  and 
the  analyzer  a  quarter- 
undulation  mica  plate  of 
the  stated  form  in  such 
a  way  that  its  long  direc- 
tion includes  45°  with 
the  arms  of  the  black 
cross.  There  then  ap- 
pear, instead  of  the  cross, 
two  black  spots;  and  if 
the  line  connecting  these 
spots  forms  a  cross  (  + ) 
with  the  long  direction 
of  the  mica,  i.e.  stands 
perpendicular  to  it,  the 
crystal  is  positive  (+), 
while  if  that  connecting 

line  is  identical  with  the  long  direction  (  — )  the  uniaxial  crystal 
is  negative  (  — ).  In  the  former  case  the  rings  are  widened  in 
those  quadrants  through  which  the  long  direction  of  the  mica 
does  not  pass;  in  the  latter  case,  in  those  it  bisects. 

The  same  mica  plate  may  serve  also  to  determine  the  sign 
of  the  double  refraction  in  the  case  of  a  biaxial  crystal  plate,  if 
this  is  cut  perpendicular  to  the  acute  bisectrix  and  therefore 
exhibits  in  convergent  light  the  two  axial  figures.  For  this 
purpose  the  crystal  plate  is  so  placed  in  the  instrument  that  its 
axial  plane  is  parallel  to  the  polarization  plane  of  one  of  the 
nicols,  and  that  accordingly  the  lemniscates  are  intersected  by 
a  black  cross.  Between  crystal  plate  and  analyzer  one  then 
inserts  the  mica  plate;  and  if  the  crystal  to  be  investigated  is 
positive,  i.e.  the  acute  bisectrix  the  vibration  direction  of  least 
light  velocity,  one  now  observes  the  interference-figure  repre- 
sented in  Fig.  105,  in  which  the  color  rings  are  widened  in  the 


210     COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

two  quadrants  through  which  the  mica  sheet  — whose  long 
direction  is  indicated  by  the  line  MM' — does  not  pass.  When 
on  the  other  hand  the  crystal  is  negative,  i.e.  its  acute  bisectrix 

the  vibration  direction 
of  greatest  light  velocity, 
then,  with  the  same  posi- 
tion of  crystal  plate  and 
of  mica,  there  appears 
the  phenomenon  Fig. 
106,  in  which  the  rings  of 
those  two  quadrants  are 
widened  that  the  long 
direction  of  the  mica 
bisects.  From  this  an- 
alogy of  the  phenom- 
ena with  those  of  uni- 


M' 


axial 


Fig.  106. 


crystals  it  is  at 
once  seen  that  likewise 
the  explanation  must 

be  analogous,  although  with  a  biaxial  crystal  the  deducing  of 
it  is  far  more  complicated.  Yet  a  simple  reflection  teaches 
that,  by  a  quarter-undulation  mica  plate,  the  interference- 
figure  of  a  crystal  plate  cut  perpendicular  to  the  obtuse  bisectrix 
(to  be  observed  only  in  a  strongly  refracting  liquid,  as  a  rule) 
must  experience  exactly  the  opposite  alteration.  Therefore  the 
obtuse  bisectrix  of  a  positive  crystal  exhibits  the  phenomenon 
Fig.  106;  that  of  a  negative,  Fig.  105.* 

REMARK.  —  It  must  be  remarked  that  by  different  optical  firms  (Steeg  and 
Reuter,  Fuess)  gypsum  lamellae  giving  the  red  of  the  first  order,  quarter-undula- 
tion mica  plates,  etc.,  are  now  supplied  in  the  reverse  orientation,  the  longer  side 
thus  being  parallel  to  the  greater  of  the-  two  light  velocities.  (See  Leiss:  Die 
op.  Instrum.,  210-212.)  If  desired  such  lamellae,  plates,  etc.,  having  the  orien- 
tation formerly  in  common  use,  and  on  which  page  198  et  seq.  are  based,  may 
of  course  still  be  obtained. 


*  For  this  reason  a  bisectrix  that,  when  investigated  with  the  quarter-undula- 
tion mica  plate,  exhibits  the  phenomenon  Fig.  105  is  pretty  generally  designated 


BEHAVIOR  OF   COMBINATIONS  OF  LIKE   CRYSTALS         21 1 

OPTICAL  BEHAVIOR  OF  COMBINATIONS  OF  DOUBLY  RE- 
FRACTING CRYSTALS  OF  THE  SAME  KIND 

From  the  two  optically  biaxial  minerals  last  mentioned,  mica 
and  gypsum,  plane-parallel  plates  of  any  desired  thickness  may 
be  very  easily  prepared,  in  virtue  of  the  highly  perfect  cleavage 
of  these  minerals  along  one  plane  (which  in  the  case  of  mica  is 
almost  exactly  perpendicular  to  the  acute  bisectrix,  in  that  of 
gypsum  parallel  to  the  optic  axial  plane) ;  and  these  plates  may 
be  grouped  in  any  way.  Now  the  behavior  of  light  when  it 
passes  through  such  a  combination  of  plates  —  a  packet,  as  it 
were,  of  mica  or  gypsum  lamellae  arranged  in  layers — is  of 
great  importance  for  the  study  of  so-called  " twinning";  i.e.  the 
regular  (but  not  parallel)  growing-together  of  several  crystals, 
which  not  infrequently  takes  place  in  such  a  way  that  it  cor- 
responds to  a  superposition  of  differently  oriented  lamellae  in 
layers.  Such  masses,  of  course,  in  their  optical  relations,  behave 
differently  from  simple  crystals.  Therefore  the  most  important 
cases  of  such  combinations  of  doubly  refracting  crystals  shall 
now  be  treated;  and  of  these  combinations  the  greater  number 
are  such  as  may  easily  be  realized  by  arranging  mica  or  gypsum 
lamellae  one  above  the  other  in  layers. 

It  is  in  general  evident,  in  the  first  place,  that  by  superposing 
two  doubly  refracting  crystal  plates  in  such  a  way  that  their 
vibration  directions  coincide  one  obtains  a  packet  having  the 
same  vibration  directions,  —  a  packet  therefore  that  in  parallel 
polarized  light  becomes  dark  in  the  same  positions  as  do  the 
single  plates.  What  phenomenon  such  a  packet  exhibits  in 

as  positive,  and  one  exhibiting  the  phenomenon  Fig.  106,  as  negative.  Yet  this 
designation  is  not  entirely  correct,  since,  as  follows  from  pp.  137-138,  with 
biaxial  crystals  the  distinction  between  "positive"  and  "negative"  refers  to  the 
greater  similarity  of  the  form  of  the  index-surface  to  that  of  a  uniaxial  positive 
or  negative  crystal;  not,  therefore,  to  a  definite  direction.  If  about  a  bisectrix  one 
observes  the  phenomenon  Fig.  105,  and  designates  that  bisectrix  as  "positive  bisec- 
trix" or  as  "bisectrix  with  positive  double  refraction",  this  signifies,  strictly  speak- 
ing, only  that  the  bisectrix  in  question  is  the  vibration  direction  of  least  light  velocity, 
and  therefore  that  the  crystal  would  be  positive  if  that  bisectrix  were  the  acute. 


212       COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

convergent  light  depends  on  the  optical  properties  of  the  two 
plates  and  on  their  thickness.  To  begin  with  a  simple  ex- 
ample, let  us  suppose  the  plates  to  be  uniaxial  and  cut  oblique 
to  the  optic  axis,  yet  so  cut  that  the  axial  figure  falls  within  the 
field  of  the  conoscope.  Then,  if  they  are  superposed  in  such  a 
way  that  their  principal  sections  are  parallel  to  each  other  and 
to  the  principal  section  of  one  of  the  crossed  nicols,  but  their 
optic  axes  inclined  to  the  normal  of  the  plate  in  opposite  direc- 
tions, one  sees,  equidistant  from  the  center  of  the  field,  the 
black  cross  and  color  rings  of  each,  and  with  rings  of  equal 
width  if  the  two  plates  are  of  equal  thickness;  visible  between 
these  two  axial  figures  are  secondary  color  stripes,  arising  by 
the  combined  action  of  the  two  crystals.*  If,  instead  of  uni- 
axial, one  takes  for  this  experiment  biaxial  plates  from  which  the 
rays  parallel  to  an  optic  axis  emerge  obliquely,  then,  when  the 
optic  axial  plane  is  parallel  to  the  principal  section  of  a  nicol, 
the  interference  phenomenon  in  convergent  light  resembles  that 
of  a  single  plate  cut  perpendicular  to  the  acute  bisectrix;  for  in 
the  field,  symmetrically  on  the  two  sides,  there  appear  two  axial 
figures — lemniscates  intersected  by  a  dark  bar.  But  this  phe- 
nomenon, which  not  infrequently  is  to  be  observed  in  combina- 
tions of  two  crystals  of  certain  substances  grown  together  in 
nature,  differs  from  the  interference-figure  of  the  single  plate  just 
referred  to,  by  the  presence  of  the  above-mentioned  secondary 
color  stripes,  — these  stripes  here  taking  the  place  of  the  second 
dark  bar,  which  there  passes  through  the  center  of  the  field  at 
right  angles  to  the  optic  axial  plane.  When  two  plates  having 

*  These  interference  phenomena  and  a  series  of  similar  ones,  produced  by 
"twin  plates"  chiefly  in  homogeneous  sodium-light,  are  contained  in  the  col- 
lection of  excellent  photographs  published  by  H.  Hauswaldt  ("  Interferenzer- 
scheinungen  an  doppeltbrechenden  Krystallplatten  in  convergenten  polarisierten 
Licht.  Photographisch  aufgenommen  von  Hans  Hauswaldt  in  Magdeburg. 
Mit  einem  Vorwort  von  Th.  Liebisch  in  Gottingen."  Magdeburg,  1902).  A 
fresh  series,  appearing  in  1904,  contains  further  examples  and,  in  addition,  pho- 
tographs of  interference  phenomena  of  biaxial  crystals  having  great  dispersion  and 
with  the  use  of  different  wave  lengths;  also  of  anomalous  crystals,  absorption 
spectra,  et  al. 


BEHAVIOR   OF   COMBINATIONS  OF  LIKE   CRYSTALS        213 


their  plane  perpendicular  to  the  acute  bisectrix  — and  for  this  pur- 
pose, as  already  mentioned,  two  cleavage- plates  of  mica  are  the 
most  suitable  —  are  so  laid,  one  upon  the  other,  that,  while  their 
vibration  directions  are  indeed  parallel,  their  optic  axial  planes 
are  crossed  at  right  angles,  one  sees  in  the  field  of  the  conoscope 
four  axial  figures,  all  equidistant  from  one  another  and  from  the 
center;  between  them  appear  hyperbolic  color  stripes,  whose 
asymptotes  form  a  black  cross  passing  through  the  center;  and 
when  the  two  axial  planes  are  in  diagonal  position  to  the  nicols, 
the  arms  of  this  cross  pass  par- 
allel to  their  principal  sections.  If 
several  such  pairs  of  crossed  mica 
plates  are  arranged  one  above  the 
other,  these  secondary  interference- 
curves  stand  out  still  more,  at  the 
expense  of  the  lemniscates;  and 
with  four  or  five  pairs  we  obtain 
the  interference  phenomenon  repre- 
sented in  Fig.  107:  of  the  color 
curves  surrounding  the  four  axial 
points  this  phenomenon  exhibits 


Fig.  107. 


only  remnants,  and  it  closely  resembles  the  interference-figure 
portrayed  in  Fig.  5  on  Plate  II  —  that  of  a  crystal  having 
crossed  axial  planes  for  different  colors.  The  center  of  this 
interference-figure  always  remains  dark,  even  when  the  combi- 
nation is  rotated  in  its  own  plane,  provided  the  single  plates  are 
of  exactly  equal  thickness;  otherwise  there  remains  a  path  dif- 
ference even  for  the  rays  that  have  passed  through  perpen- 
dicularly, and  on  rotation  a  color  arises  at  the  center. 

A  quite  different  interference  phenomenon  (observed  first  by 
Norrenberg)  is  obtained,  on  the  other  hand,  if  the  single  mica 
lamellae  arranged  crosswise  in  layers  are  so  thin  that  in  any  one 
of  them  the  two  rays  arising  by  the  double  refraction  acquire 
less  than  one  wave  length  difference  of  path.  When  the  prin- 
cipal sections  of  the  mica  coincide  respectively  with  the  two 


214      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

crossed  nicols  of  the  instrument,  we  observe  an  interference- 
figure  which  agrees  absolutely  with  that  of  a  plate  cut  perpen- 
dicular to  the  optic  axis  from  a  uniaxial  crystal,  consisting 
therefore  of  circular  color  rings  intersected  by  a  dark  cross. 
That  such  a  packet  composed  of  mica  lamellae  of  exactly  equal 
thinness  must,  as  in  the  foregoing  case,  behave,  for  rays  passing 
through  perpendicularly,  like  a  uniaxial  plate  cut  normal  to  the 
optic  axis  (i.e.  in  parallel  light  appear  singly  refracting),  is 
clear.  For  the  small  difference  of  path  produced  by  the  double 
refraction  in  one  of  the  lamellae  is  done  away  with  by  the  next, 
in  which  the  two  vibration  directions  are  interchanged;  and 
hence,  the  sum  of  all  the  positive  differences  of  path  in  the  one 
half  of  the  lamellae  thus  being  entirely  counterbalanced  by  the 
sum  of  all  the  opposite  differences  of  path  arising  in  the  other 
half,  the  rays  of  this  direction  suffer  no  double  refraction. 
Different,  however,  is  in  general  the  behavior  of  rays  that  pass 
through  such  a  combination  obliquely:  their  transmission  direc- 
tion in  the  one  kind  of  mica  plates  has  a  different  orientation 
from  that  in  the  other  kind,  crossed  as  regards  the  former;  so 
that,  in  the  two  systems  of  mica  lamellae,  such  rays  experience 
a  different  double  refraction  and  in  the  end  emerge  with  a 
difference  of  path,  this  producing  a  definite  color.  Thus,  in 
convergent  light,  isochromatic  curves  must  appear,  which  in  the 
case  of  a  smaller  number  of  plates  being  superposed  have  the 
form  of  hyperbolas,  but  which  pass  over  into  concentric  circles, 
on  the  other  hand,  when  the  number  of  plates  is  very  large  and 
their  thickness  very  slight.  If  the  single  lamellae  of  the  one 
system,  while  themselves  all  of  equal  thinness,  are  thicker  or 
thinner  than  those  of  the  other  system,  crossed  as  regards  the 
former,  then  does  the  whole  packet  — and  again  the  more  per- 
fectly, the  larger  the  number  and  the  less  the  thickness  of  the 
lamellae  —  operate  like  a  plate  cut  perpendicular  to  the  acute 
bisectrix  from  a  single  biaxial  crystal,  a  crystal  however  whose 
axial  angle  were  the  smaller,  the  less  the  lamellae  of  the  two 
svstems  differed  in  thickness. 


BEHAVIOR  OF   COMBINATIONS   OF  LIKE   CRYSTALS        215 

These  phenomena  become  more  complicated  when  the  vibra- 
tion directions  of  the  combined  plates  do  not  coincide;  for  ex- 
ample, when  the  two  gypsum  or  mica  plates  of  equal  thickness 
are  superposed  not  parallel  or  at  right  angles  but  crossed 
obliquely.  From  such  a  combination,  supposing  it  to  be  illumi- 
nated with  parallel  polarized  light,  there  in  general  then  emerges 
light  that  is  elliptically  polarized;  and,  since  a  light  ray  of  this 
kind  (see  p.  20)  corresponds  to  two  mutually  perpendicular 
vibrations  of  unequal  intensity,  and  since  consequently  on  the 
resolution  in  a  Nicol  prism  only  a  portion  of  these  two  vibra- 
tions can  be  annihilated,  the  emerging  light  is  not  extinguished 
by  the  analyzer  in  any  position  of  the  same.  If  one  rotates  the 
combination  itself,  between  crossed  nicols,  then,  although  it 
does  exhibit  a  decrease  and  increase  of  brightness,  correspond- 
ing to  the  orientation  of  the  major  and  of  the  minor  axis  of  the 
elliptical  path  of  the  ether  vibrations,  yet  it  never  becomes  quite 
dark.  In  case  the  vibration  directions  of  the  two  equally  thick 
plates  form  45°  with  each  other,  the  axes  of  the  ellipse  are  of 
equal  length:  the  light  emerging  from  the  combination  is  cir- 
cularly polarized  (see  p.  20)  and  therefore,  on  the  rotation 
between  crossed  nicols,  suffers  no  variation  in  intensity.  As 
with  the  plates  parallel  and  at  right  angles,  so  too  when  they 
are  crossed  obliquely  do  the  phenomena  in  convergent  light 
depend  on  the  nature  and  thickness  of  the  plates:  provided 
their  thickness  is  not  too  slight,  and  if  from  them  there  emerge 
rays  parallel  to  the  optic  axis,  one  observes,  as  with  plates  com- 
bined at  right  angles,  the  isochromatic  curves  of  both  systems  at 
once,  in  the  relative  orientation  corresponding  to  the  crossing- 
angle;  and  between  these  curves  there  appear  color  curves 
whose  form,  likewise,  depends  on  the  angle  mentioned. 

As  in  the  case  first  considered,  of  the  crystal  plates  being 
crossed  at  right  angles,  so  too  when  they  cross  at  any  oblique 
angle  does  the  optical  behavior  of  the  packet  pass  over  into 
that  of  a  homogeneous  crystal  if  the  lamellae  are  taken  thin 
enough  and  their  number  sufficiently  large.  When  very  thin 


21 6      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

cleavage  sheets  of  mica  are  arranged  alternately  at  an  oblique 
angle  above  one  another,  the  resulting  packet  behaves,  optically, 
like  a  single  biaxial  crystal  whose  optic  axial  plane  bisects  the 
acute  angle  formed  by  the  common  axial  plane  of  the  single 
lamellae  of  the  one  system  with  that  of  the  lamellae  of  the  other 
system,  and  whose  optic  axial  angle  is  smaller  than  that  of  the 
mica  used.  Let,  for  example,  in  Fig.  1080,  the  long  direction 
of  the  rectangular  mica  plates  marked  i  and  2  be  the  trace  of 
their  optic  axial  plane;  and  let  each  of  these  two  rectangles  repre- 
sent a  number  of  parallel  mica  lamellae  lying  one  above  the 
other,  so  that  in  the  little  triangle  under  the  central  hexagon,  R, 
there  lie  the  two  kinds  of  lamellae,  crossed  at  60°,  one  above  the 
other  in  alternate  layers.  Investigated  in  parallel  polarized 
light,  this  triangle  then  exhibits  the  exit  of  two  plane-polarized 
rays  whose  vibration  directions  are  oriented  horizontal  and 
vertical  respectively,  thus  bisecting  the  acute  and  the  obtuse 
angle  of  the  two  systems  of  mica  lamellae;  if  by  rotating  the 
combination  these  directions  are  brought  into  coincidence  with 
the  principal  sections  of  the  nicols,  the  triangle  appears  quite 
dark,  like  a  homogeneous  doubly  refracting  crystal.  In  the 
conoscope  it  exhibits  the  normal  axial  figure  of  a  biaxial  crystal 
having  its  optic  axes  oriented  as  mentioned  at. the  top  of  the 
page. 

A  biaxial  crystal  plate  that  is  cut  perpendicular  to  the  acute 
bisectrix,  and  in  which  the  principal  vibration  directions  for 
different  colors  are  not  dispersed,  will  obviously,  when  rotated 
1 80°  about  the  bisectrix,  pass  over  into  a  position  where  it  must 
exhibit  exactly  the  same  behavior  toward  light  as  before;  be- 
cause then,  in  consequence  of  the  symmetry  throughout  with 
reference  to  the  principal  optic  sections,  equivalent  directions 
must  arrive  at  the  same  orientation.  With  mica  these  con- 
ditions are,  to  be  sure,  not  fulfilled  with  absolute  exactness,  yet 
they  are  so  nearly,  that  the  small  deviations  have  no  essential 
influence  on  the  resulting  phenomena.  Therefore,  if  upon  a 
thin  rectangular  mica  lamella  whose  long  direction  corresponds 


BEHAVIOR  OF   COMBINATIONS   OF   LIKE   CRYSTALS        217 

to  the  optic  axial  plane  one  lays  a  second,  at  an  acute  angle 
dividing  into  180°  without  a  remainder  (e.g.  60°  or  45°),  and  if 
upon  this  one  lays  a  third,  rotated  by  that  same  angle  relatively 
to  the  second,  and  so  on,  then  in  the  first  case  (60°)  the  fourth 
lamella,  in  the  second  case  (45°)  the  fifth  lamella,  is  parallel  to 
the  first ;  in  other  words,  after  every  three  or  every  four  plates  the 
same  optical  orientation  will  recur;  a  column  arises,  built  up  of 
similar  packets  of  three,  four,  etc.,  lamellae.  But,  as  is  seen 


3 

\     / 

A 

L 

X 

Fig.  1 08 


from  a  glance  at  the  above  figure,  which  is  based  on  a  rotation 
angle  of  60°,  such  packets  may  be  combined  in  two  ways;  viz. 
either  so  that  the  single  lamellae  form  winding  stairs  ascending 
from  left  to  right,  as  represented  in  Fig.  io8a,  or  so  that  they 
form  stairs  ascending  from  right  to  left,  as  shown  in  Fig.  1086. 
Now  in  the  central  hexagon  (R  and  L),  where  the  plates  of  all 
three  orientations  lie  in  equal  number  above  one  another,  a 
column  of  such  packets  —  called,  after  its  inventor,  Reusch's 
mica  combination  —  behaves  differently  according  to  the  sense 
in  which  the  packets  themselves  are  built  up.  Between  crossed 
nicols,  in  parallel  light,  the  hexagon  appears  bright  in  every 
position;  but  if  the  analyzer  is  rotated  a  certain  angle  we 
obtain  complete  extinction  of  the  emerging  light,  provided  it  is 
monochromatic;  if  white  light  is  employed,  then,  as  the  upper 
nicol  of  the  orthoscope  is  rotated,  changing  colors  appear.  These 
phenomena  are  explained  by  the  fact  that,  while  rays  passing 


2l8      COMBINATIONS  OF  DOUBLY  REFRACTING  CRYSTALS 

through  the  combination  perpendicularly  are  indeed  not  doubly 
refracted,  their  vibration  direction  (or,  what  means  the  same,  their 
polarization  plane)  on  the  other  hand  experiences  a  rotation;  and 
that  for  different  colors  this  rotation  is  unequal,  so  that  in  white 
light,  by  the  analyzer  in  any  single  position,  only  one  color  can 
be  extinguished.  Now  in  a  mica  combination  such  as  that  of 
Fig.  loSa  this  rotation  of  the  polarization  plane  is  to  the  right, 
i.e.  clockwise  ("  right ",  or  "dextro",  rotation);  in  that  of 
1086,  to  the  left  ("left",  or  "  levo  ",  rotation),  but  of  exactly 
the  same  amount  if  the  two  combinations  are  composed  of  the 
same  number  of  equally  thin  lamellae,  the  one  thus  presenting 
the  exact  reflected  image  of  the  other.  The  same  phenomenon 
is  exhibited  also  by  certain  natural  crystals;  namely,  by  crystals 
for  which,  from  their  opposite  crystal  form,  like  an  object  and 
its  reflected  image,  an  analogous  structure  must  be  inferred, 
consisting  in  a  dextro  or  a  levo  spiral  arrangement  of  the 
smallest  particles.  The  description  of  the  optical  behavior  of 
such  crystals  (called  "  crystals  with  optical  rotatory  power ", 
or  rotatory  crystals)  will  form  the  subject  of  the  next  sec- 
tion. —  In  convergent  light  the  Reusch  mica  combinations  * 
exhibit  in  general  the  behavior  of  optically  uniaxial  crystal 
plates,  except  that  the  black  cross  traversing  the  color  rings  is 
lighted  up  at  the  center;  if  white  light  is  employed,  a  color  ap- 
pears there,  and  on  rotation  of  the  analyzer  this  color  changes. 
Therefore  the  phenomena  described  in  the  following  section 
for  the  rotatory  uniaxial  crystals,  especially  for  those  to  be  ob- 
served in  the  combination  of  a  right,  or  dextro  (i.e.  dextro- 
rotatory), with  a  left,  or  levo  (i.e.  levo-rotatory),  crystal,  may 
be  produced  likewise  with  dextro-  and  with  levo-rotatory  mica 
combinations. 

From  the  behavior  of  the  mica  combinations  mentioned  on 

*  Reusch  mica  combinations  of  excellent  workmanship  are  supplied  by 
the  firm  of  Steeg  and  Reuter  in  Homburg,  being  composed  of  thirty  mica 
lamellae  so  thin  that  singly  they  produce  a  path  difference  of  only  £  X.  The  rota- 
tory power  of  such  a  preparation  is  equal  to  that  of  a  quartz  plate  8  mm.  thick. 
(See  following  section.) 


BEHAVIOR  OF   COMBINATIONS   OF  LIKE   CRYSTALS         219 

pages  213-214  the  following  conclusions  may  be  drawn:  First,  a 
plate  cut  normal  to  the  acute  bisectrix  from  a  "  twin  crystal  " 
(see  p.  21 1 )  that  is  built  up  of  biaxial  lamellae  crossed  in  mu- 
tually perpendicular  orientation  but  having  different  size  and 
thickness,  must,  at  points  where  the  light  passes  through  layers 
only  of  one  orientation,  exhibit  the  biaxial  interference-figure 
with  normal  axial  angle;  second,  at  points  where  between  the 
layers  of  this  orientation  there  are  interlaid  fine  lamellae  of  the 
other,  the  interference-figure  corresponds  to  a  smaller  axial  angle, 
and  when  the  lamellae  of  the  latter  orientation  predominate 
the  plane  of  this  angle  stands  perpendicular  to  tha"t  of  the 
axial  angle  first  mentioned;  third  and  last,  at  points  where  the 
orientations  are  evenly  balanced  the  axial  angle  becomes  zero, 
and  the  interference-figure  of  a  uniaxial  crystal  appears.  In 
like  manner  an  optically  biaxial  crystal  built  up  of  twinned 
lamellae  whose  axial  planes  form  oblique  angles  with  one  an- 
other will,  according  to  the  way  in  which  it  is  built  up,  exhibit 
at  different  points  a  different  axial  angle  and  a  different  orientation 
of  the  optic  axial  plane;  under  some  circumstances  even  uni- 
axial phenomena  may  appear,  combined  with  rotation  of  the 
polarization  plane.  Hence,  if  in  such  a  composite  crystal  the 
lamellae  are  so  thin  that  they  themselves  escape  observation  in 
the  microscope,  then  how  it  is  built  up  can  be  concluded  only 
from  the  aggregate  effect  of  the  lamellae  upon  the  polarized 
light  passing  through  them:  when  the  structure  is  exactly  the 
same  at  all  points,  the  optical  behavior  must  be  that,  corre- 
sponding to  this  aggregate  effect,  of  a  homogeneous  crystal;  but 
when  the  different  parts  of  the  crystal  differ  in  their  structure  in 
the  manner  just  stated,  the  crystal  appears  heterogeneous,  ex- 
hibiting at  different  points  a  different  optical  behavior.  Such 
crystals  are  said  to  be  optically  anomalous.  (Cf.  pp.  279- 
282.)  Properly  speaking,  however,  it  is  here  no  question  of  an 
"  anomaly";  for  the  behavior  of  such  a  composite  mass  must 
necessarily  follow  regularly  from  the  nature  of  the  component 
lamellae.  Mallard,  starting  out  from  the  undulatory  theory  of 


220  ROTATION   OF  THE  POLARIZATION   PLANE 

light,  deduced  quite  generally  the  changes  that  a  plane-polar- 
ized light  ray  must  suffer  in  passing  through  packets  of  thin 
doubly  refracting  crystal  lamellae  oriented  at  will,  and  thereby 
he  supplied  a  complete  theoretical  explanation  of  all  the  phe- 
nomena considered  in  the  foregoing.  On  the  basis  of  this  theory 
there  follow,  from  the  optical  constants  of  the  single  lamellae  of 
such  a  packet,  those  of  the  packet  itself;  so  that  in  cases  where 
a  substance  occurs  both  homogeneous,  i.e.  in  single  crys- 
tals, and  in  crystals  apparently  homogeneous  but  really  built 
up  of  fine  twinned  lamellae  (polysynthetic),*  the  optical  con- 
stants of  these  lamellae  may  be  calculated  from  those  of  the 
homogeneous  crystal. 

ROTATION  OF  THE  POLARIZATION  PLANE  OF 
LIGHT  IN  CRYSTALS 

Independently  of  whether  they  are  doubly  refracting  or  not, 
the  crystals  of  a  number  of  substances  exhibit  a  property  that 
belongs  also  to  certain  liquids,  as  oil  of  turpentine,  sugar  solu- 
tion, and  others,  —  the  property,  namely,  of  rotating  the  polar- 
ization plane  of  a  plane-polarized  light  ray  passing  through  it; 
the  rotation  amounts  to  a  definite  angle,  this  depending  on 
the  color  of  the  light  and  on  the  nature  of  the  crystal.  Such 
substances  are  said  to  be  optically  active. 

The  cause  of  this  phenomenon  is  a  peculiar  kind  of  double 
refraction,  quite  different  from  the  ordinary,  within  these  crys- 
tals: the  entering  plane-polarized,  i.e.  rectilinear,  vibration  is  in 
such  a  crystal  resolved  into  two  circular  vibrations,  of  which  the 
one  takes  place  clockwise,  the  other  in  the  opposite  sense;  these 
two  "  circularly  polarized  "  (see  p.  20)  rays  of  light  are  trans- 
mitted in  the  crystal  with  different  velocity  and  therefore  leave 
it  with  a  difference  of  path,  which  depends  on  the  thickness  of 

*  Such  substances,  which  appear  to  be  differently  crystallized,  are  designated 
as  "polysymmetric".  (See  the  author's  "An  Introduction  to  Chemical  Crystallog- 
raphy". English  by  Hugh  Marshall,  Edinburgh  and  New  York,  1906,  p.  7.) 


ROTATION   OF  THE   POLARIZATION   PLANE  221 

the  crystal.  Now  the  theory  teaches  that  two  circular  vibra- 
tions, one  retarded  as  compared  with  the  other,  must  combine  to 
form  a  rectilinear,  i.e.  plane-polarized,  vibration  which  as  com- 
pared with  the  original  vibrations  is  rotated  an  angle  depend- 
ing on  the  amount  of  this  retardation.  Since,  then,  in  such 
a  crystal  the  arising  path  difference  of  the  two  circular  rays 
is  different,  according  to  their  vibration-period,  — being  the 
greater,  the  greater  the  refrangibility  of  the  color  in  question, — • 
the  rotation  of  the  vibration  plane  produced  by  a  plate  cut  from 
such  a  crystal  is  least  for  red  rays,  greater  for  yellow,  green, 
blue,  and  greatest  for  violet;  so  the  several  vibration  directions, 
parallel  before  entrance,  of  the  component  rays  of  the  incident 
white  light  are  by  such  a  crystal  dispersed.  This  dispersion 
takes  place  after  a  law  similar  to  that  holding  good  with  the 
dispersion  produced  by  a  refracting  prism.  (Cf.  p.  46.)  If 
A  and  B  represent  the  constants  that  apply  to  a  certain  body, 
and  a  the  rotation  angle  for  a  plate  one  millimeter  thick,  then 
according  to  Boltzmann  the  relation  between  rotation  angle  and 
wave  length  (in  air)  may  be  expressed  as  follows: 

A      B 

a  --  r  +  *  + 

From  the  measured  rotation  of  the  polarization  plane  for  at 
least  two  colors  of  known  wave  length  we  may  therefore  cal- 
culate that  for  the  remaining  colors,  just  as  with  the  above- 
mentioned  dispersion  formula  for  refraction  of  light.  Thus,  for 
example,  by  plates  of  sodium  chlorate  (according  to  Guye)  and 
quartz  (according  to  Soret  and  Sarasin)  i  mm.  thick  the  vibra- 
tion plane  of  the  light  corresponding  to  the  several  Fraunhofer's 
lines  indicated  by  the  letters  at  the  top  of  the  following  table  is 
rotated  the  angles  specified  below  the  letters: 

B         C         D          E         F         G         H 

Sodium  Chlorate  :  a  =  2°  27  2°.5o  3°.i3  3^94  4°.67  6°.oo  7°.i7 
Quartz  i5°.75  i7°.3i  2i°.7i  27^54  320.76  42°.59  S*°-*9 

As  one  sees  from  these  examples,  the  specific  rotatory  power 


222  ROTATION   OF  THE   POLARIZATION   PLANE 

of  different  substances  is  very  different.  Further,  since  the  rela- 
tive retardation  of  the  two  circular  vibrations  arising  in  such  a 
crystal  must  be  twice  as  great  if  the  rays  have  twice  the  distance 
to  travel  in  the  crystal,  the  rotation  must  increase  in  proportion  to 
the  thickness  of  the  plate;  thus,  for  a  plate  of  double  the  thick- 
ness the  rotation  angle  is  double  the  value  given,  and  so  on. 

Now  of  each  of  the  crystallized  substances  here  coming  into 
consideration  there  are  two  optically  opposite  modifications, 
with  equal  but  opposite  rotatory  power,  this  being  in  conse- 
quence of  the  circumstance  that  in  the  crystal  of  the  one  modi- 
fication the  dextro-rotary  (clockwise)  vibration  has  the  greater 
transmission  velocity,  in  those  of  the  other  the  levo-rotary; 
crystals  of  the  first  sort  rotate  the  polarization  plane  from  left  to 
right,  those  of  the  second  sort  — in  which,  with  the  same  plate 
thickness,  the  path  difference  is  equal  in  amount  but  opposite 
in  sense  —  just  as  far  from  right  to  left. 

Optical  activity  has  been  observed,  up  to  the  present  time, 
in  the  following  singly  refracting  bodies: 

Sodium  chlorate 

Sodium  bromate 

Sodium  sulphantimonate  (Schlippe's  salt) 

Uranyl  sodium  acetate 

Active  amylamin  alum 

Conin  alum 

In  these  crystals  the  phenomena  resulting  from  the  rotation 
of  the  polarization  plane  are  exhibited  in  the  same  way  in  every 
direction;  that  is,  their  specific  rotatory  power  is  equal  for  all 
directions,  depending  only  on  the  color  of  the  light.  Plates  of 
equal  thickness  all  behave  alike,  in  whatever  direction  they  be 
cut  from  such  a  crystal.  But  with  optically  uniaxial  crystals  it 
is  otherwise;  of  these  a  larger  number,  as  follows,  have  been 
recognized  as  possessing  optical  rotatory  power: 

Quartz  (silica,  silicon  dioxid) 

Cinnabar  (mercuric  sulphid) 

Potassium  lithium  sulphate  and  seleniate 


ROTATION    OF   THE    POLARIZATION   PLANE  223 

Rubidium  lithium  sulphate 

Ammonium  lithium  sulphate 

Potassium  lithium  chromate  sulphate 

Potassium  dithionate  and  the  corresponding  rubidium  salt 

Calcium  dithionate 

Strontium  dithionate 

Lead  dithionate 

Sodium  periodate 

Guanidin  carbonate 

Rubidium  and  caesium  tartrates 

Ethylenediamin  sulphate 

Benzil  (diphenyldiketone) 

Di-acetylphenolphthalein 

Common  camphor  and  matico  camphor 

Strychnin  sulphate 

Sulphobenzoltrisulphid 

The  laws  of  circular  double  refraction  and  of  the  rotation 
of  the  polarization  plane  resulting  from  it  were  discovered  by 
Arago  first  in  quartz,  the  most  important  of  the  series  of  opti- 
cally uniaxial  crystals  just  enumerated,  and  more  closely  investi- 
gated chiefly  by  Biot  and  Fresnel;  they  are  as  follows:  — 

In  the  optic-axial  direction,  since  there  is  here  no  double  re- 
fraction of  the  ordinary  kind,  only  those  phenomena  arise  that 
are  dependent  solely  on  the  circular  double  refraction.  For 
example,  a  quartz  plate  cut  perpendicular  to  the  axis  exhibits, 
when  illuminated  only  by  plane-polarized  rays  falling  normal  to 
itj  the  same  behavior  as  a  plate  of  corresponding  thickness  cut  in 
any  direction  from  a  singly  refracting  rotatory  crystal.  But,  so 
soon  as  the  entering  light  rays  are  inclined  to  the  optic  axis, 
there  is  added  to  the  path  difference  resulting  from  the  circular 
double  refraction  that  produced  by  the  ordinary  double  refrac- 
tion; and  this  latter  path  difference  grows  larger  with  increas- 
ing inclination  of  the  rays  to  the  axis.  The  behavior  hereby 
effected,  of  an  optically  active  uniaxial  crystal  toward  light  rays, 
follows  from  the  theory,  developed  by  Gouy  and  in  a  somewhat 


224  ROTATION    OF  THE   POLARIZATION   PLANE 

different  form  by  Wiener,  of  the  joint  action  of  circular  and 
of  linear  (plane)  double  refraction;  and  this  theory  rests  on  the 
presupposition  that  the  two  differences  of  path,  the  one  pro- 
duced by  the  circular  doublp  refraction,  the  other  by  the  ordi- 
nary, are  simply  superposed.  Experimental  investigations  made 
with  quartz  have  quite  borne  out  the  conclusions  deduced  from 
this  theory.  These  investigations  have  proved  especially  that, 
just  as  in  the  optically  active  singly  refracting  crystals,  with 
equal  plate  thickness  and  the  same  color  of  the  light  rays  the 
former  of  these  two  differences  of  path  retains  the  same  value  in 
all  directions;  but  also  that  by  reason  of  the  ordinary  double 
refraction  the  effect  of  the  circular  is,  with  increasing  inclination 
to  the  axis,  diminished  more  and  more,  until  finally  it  is  wholly 
annihilated.  This  diminishing  effect  of  the  circular  double  re- 
fraction is  explained  as  follows:  The  two  circular  vibrations 
transmitted  parallel  to  the  axis  are  transformed,  when  the  rays 
include  a  small  angle  with  this  direction,  into  two  elliptical  vibra- 
tions, whose  respective  paths  are  similarly  shaped  but  differently 
oriented,  and  traveled  by  the  ether  particles  in  the  opposite 
sense;  and  these  two  paths,  which  in  the  beginning  have  nearly 
equal  axes,  always  assume  a  more  elongated  form  as  the  incli- 
nation of  the  rays  to  the  optic  axis  becomes  greater.  When  this 
inclination  attains  a  certain  value  the  minor  axis  of  the  ellipse 
is  infinitesimal  as  compared  with  the  major,  and  the  vibration 
of  the  ether  particles  no  longer  differs  from  a  rectilinear,  i.e. 
plane-polarized,  vibration.  A  plane-polarized  ray  of  light  en- 
tering the  crystal  is  then  resolved  into  two  rays,  likewise  piane- 
polarized,  and  vibrating  perpendicularly  to  each  other,  whose 
polarization  plane  no  longer  exhibits  rotation;  in  other  words, 
the  crystal  behaves  like  an  optically  uniaxial  crystal  without 
optical  rotatory  power. 

From  the  foregoing  it  follows  then,  what  phenomena  must 
be  exhibited  by  a  plate  cut  from  quartz  perpendicular  to  the 
axis.  In  the  orthoscope,  if  illuminated  with  monochromatic 
light,  beftveen  crossed  nicols  it  appears  bright,  and  in  order  to 


ROTATION   OF  THE   POLARIZATION   PLANE  225 

annihilate  the  light  the  analyzer  must  be  rotated  a  definite 
angle,  — in  the  case  of  a  dextro  (i.e.  dextro-rotatory)  crystal 
clockwise,  in  the  case  of  a  levo-crystal  in  the  opposite  sense. 
This  rotation  angle  proves  to  be  the  larger,  the  less  the  wave 
length  of  the  light  used,  and  it  increases  for  the  same  color  in 
proportion  to  the  thickness  of  the  plate.  With  a  plate  one  milli- 
meter thick  the  rotation  angle  for  the  different  colors  amounts, 
according  to  Biot,  to  the  following  values  (the  more  exact 
values  for  the  several  Fraunhofer's  lines  have  already  been  given 
on  p.  221) : 

Extreme  red .  1 7°-5 

Medium  red i9°.o 

"        orange 21°. 4 

"        yellow 24°.o 

"        green 2j°.S 

"        blue 32^.3 

"        indigo 36°.  i 

"        violet 4o°.8 

Extreme  violet 44°-i 

It  is  because  of  the  great  diversity  in  the  amount  of  rotation 
which  accordingly  is  experienced  by  the  vibrations  of  different 
colors,  that  in  white  light  we  are  unable  with  the  analyzer  in  any 
position  to  make  the  plate  dark:  the  rays  of  different  colors 
vibrate  in  entirely  different  planes,  and  therefore  the  nicol  can- 
not extinguish  all  of  them  at  once.  If  N'N'  (Fig.  109,  p.  226) 
be  the  vibration  plane  of  the  parallel  rays  of  white  light  entering 
a  dextro-rotatory  quartz  plate  of  about  three  millimeters  thick- 
ness, the  red  rays  contained  therein  will,  in  the  quartz,  suffer 
the  least  rotation;  their  vibration  direction  on  emergence  will  be 
about  pp,  that  of  the  yellow  rays  ^,  of  the  green  7-7-,  of  the  blue 
/?/?,  of  the  violet  uu.  Supposing  the  analyzer  to  stand  perpen- 
dicular to  the  polarizer,  the  vibration  plane  of  the  light  pass- 
ing through  the  former  thus  being  parallel  to  N"N",  then  of 
the  rays  emerging  from  the  quartz  the  analyzer  will  transmit 
unweakened  only  those  that  vibrate  parallel  to  N"N"\  these 
are  the  green.  All  rays  of  the  other  colors  are  resolved  in  the 


226 


ROTATION    OF   THE   POLARIZATION   PLANE 


nicol  into  two  components,  one  of  these  being  annihilated;  and 
this  latter  component  is  the  greater,  the  larger  the  angle  its 
vibration  direction  forms  with  N"N".  Thus  all  colors  except  the 
green  are  weakened,  and  weakened  the  more,  the  larger  the  angle 
their  vibration  direction  forms  with  that,  N"N",  of  the  analyzer. 
The  aggregate  impression  of  all  these  colors  after  their  reduction 
to  one  polarization  plane  can  accordingly  no  longer  be  white,  but 
must  be  a  composite  color,  — one  in  which  green  predominates 

N' 


•y- 


-s* 


N' 
Fig.  109. 

over  the  other  colors.  Observed  in  white  light,  between  crossed 
nicols  such  a  plate  must  therefore  appear  green.  In  this  phe- 
nomenon, by  a  rotation  of  the  plate  in  its  own  plane  not  the 
least  alteration  will  be  produced,  since  herewith,  as  we  know,  the 
direction  in  which  the  rays  traverse  the  plate,  and  hence  also  the 
rotation  of  their  polarization  plane,  always  remains  the  same.  But 
in  the  case  of  a  nicol  being  rotated  from  its  position  relative  to 
the  other  nicol  it  is  otherwise.  For  example,  if  from  the  posi- 
tion N"N"  (Fig.  109)  we  rotate  the  analyzer  from  left  to  right 
(as  indicated  by  the  arrow  in  the  figure),  its  vibration  direction 
becomes  parallel  to  that  of  the  blue  light;  consequently,  in  the 


ROTATION    OF   THE   POLARIZATION    PLANE 


227 


arising  mixture  this  color  has  its  maximum  intensity,  the  others 
being  weakened,  and  the  plate  will  appear  blue.  With  further 
rotation  in  the  same  direction,  N"N"  coincides  with  the  vibration 
direction  uu  of  the  violet  light,  and  the  plate  appears  violet.  If 
we  rotate  beyond  N'N',  until  N"N"  becomes  parallel  to  pp, 
the  plate  is  colored  with  a  composite  color  in  which  red  predom- 
inates; so  we  see  red.  On  further  rotation  yellow  appears, 
then  green,  blue,  violet.  Accordingly,  if  we  rotate  the  analyzer 

N' 


Fig.  no. 

from  left  to  right  the  plate  always  appears  colored,  but  its 
coloring  changes  in  such  a  way  that  the  different  colors  of  the 
spectrum  appear  in  the  sequence  of  their  refrangibility  (from  the 
least  refrangible,  i.e.  red,  to  the  violet,  which  is  the  most  strongly 
refracted).  One  easily  sees  that  with  reversed  rotation  the  color 
sequence  is  the  opposite. 

Now  if  instead  of  the  dextro-quartz  plate  we  take  an  equally 
thick  plate  of  few-quartz,  and  if  N'N'  (Fig.  no)  again  be  the 
vibration  direction  of  the  white  light  entering  the  same,  the 
vibration  directions  of  the  red,  yellow,  green,  blue,  and  violet 
rays  after  their  exit  from  the  plate  will  here  be  pp,  tt,  77, 


228  ROTATION  OF  THE   POLARIZATION   PLANE 

/?/?,  L»U.  As  in  the  case  of  the  dextro-quartz,  therefore,  here  too 
there  is  no  position  of  the  nicols  in  which  darkness  can  arise: 
the  plate  always  exhibits  a  composite  color.  With  the  analyzer 
in  crossed  position  this  plate,  like  the  previous  one,  when  its 
vibration  direction  is  AT""  AT",  appears  green;  here,  however,  when 
the  upper  nicol  is  turned  around  from  left  to  right  the  first  color 
to  appear  is  yellow,  then  red,  and  so  on:  with  the  same  nicol 
rotation  by  which  in  using  a  dextro-quartz  plate  we  obtained 
the  colors  in  the  sequence  of  their  refrangibility,  they  appear  in 
the  case  of  a  levo-plate  in  the  reverse  sequence;  that  is,  in  order 
to  obtain  the  same  color  sequence  we  must  rotate  in  the  opposite 
direction  (as  indicated  by  the  arrow). 

Hence  it  follows  that  a  plate  of  dextro-  and  one  of  levo-quartz 
brought  between  two  nicols  of  definite  position  relative  to  each 
other,  as  crossed  perpendicularly,  exhibit  the  same  color  but 
differ  in  this  particular:  that  while  in  the  case  of  the  dextro-pl&te 
the  colors  appear  in  the  sequence  red,  orange,  yellow,  green, 
blue,  violet  with  dextro  (clockwise)  rotation  of  the  analyzer, 
when  the  plate  is  few-rotatory  the  same  sequence  appears  with 
the  opposite  rotation  of  that  nicol. 

If  we  lay  a  thicker  quartz  plate  between  two  crossed  nicols, 
it  must  exhibit  a  different  color  from  the  previous  one  observed 
between  the  same  nicols:  proportionately  to  its  thickness,  this 
plate  produces  more  rotation;  so  that  on  emerging  from  it  the 
vibration  directions  of  the  different  colors  have  a  different  (more 
rotated)  position,  and  in  consequence  the  color  extinguished  by 
the  analyzer  is  not  the  same  as  before.  With  a  plate  thickness  of 
3.75  mm.,  as  may  easily  be  seen  from  the  numerical  values  given 
on  page  225,  the  red  vibrations  are  rotated  more  than  60°,  the 
green  over  100°,  the  violet  about  180°;  when  the  nicols  are 
parallel  such  a  plate  appears  purplish  violet,  and  this  color 
(teinte  sensible,  teinte  de  passage)  goes  pretty  quickly  over  into 
red  or  blue  if  the  analyzer  is  rotated  in  either  direction.  Such  a 
plate,  a  so-called  Biot's  quartz  plate,  is  sometimes  employed  in 
microscopical  investigations  instead  of  the  gypsum  plate  exhibit- 


ROTATION   OF  THE    POLARIZATION   PLANE  2 29 

ing  the  red  of  the  first  order  (see  p.  200) ;  but  the  latter  is  to  be 
preferred,  on  account  of  its  being  more  sensitive.  On  the  other 
hand,  for  measuring  small  rotations  of  the-  polarization  plane  a 
very  suitable  contrivance  is  a  double  plate  consisting  half  of 
dextro-  and  half  of  levo-quartz,  of  3.75  mm.  thickness,  which 
between  two  nicols  exactly  parallel  or  crossed  exactly  at  right 
angles  exhibits  the  same  color  shade  in  both  halves:  if  the 
polarizer  or,  through  inserting  a  feebly  rotatory  substance,  the 
vibration  emerging  from  the  polarizer  is  rotated  a  small  angle, 
the  two  halves  of  the  double  plate  are  immediately  colored 
differently  from  each  other,  and  the  analyzer  must  be  rotated 
that  same  angle  in  order  to  make  the  color  identical  in  the  two 
halves.  Quartz  plates  of  considerably  greater  thickness  exhibit 
less  vivid  colors;  with  a  thickness  of  20  mm.,  for  example,  the 
red  rays  are  rotated  2  X  180°,  the  yellowish  green  3  X  180°,  the 
indigo-blue  4  X  180°;  so  these  rays  all  vibrate  parallel  to  one 
another,  and  in  the  analyzer  they  are  all  extinguished  simulta- 
neously. With  still  greater  thickness  the  same  applies  to  still 
more  colors  of  the  spectrum;  in  other  words,  there  appears  in 
greater  and  greater  perfection  the  white  of  a  higher  order.  But 
likewise  are  the  colors  less  vivid  when  the  crystal  plate  em- 
ployed is  very  thin;  i.e.  the  rotation  of  the  polarization  plane 
effected  by  it  very  slight.  A  quartz  plate  of  o.i  mm.  thick- 
ness, for  example,  rotates  the  plane  of  the  vibrations  for  the 
outermost  colors  only  i°.7  and  4°. 4  respectively;  and  these 
colors  can  therefore,  by  means  of  a  small  rotation  of  the  ana- 
lyzer, be  extinguished  almost  simultaneously,  while  with  the 
nicols  crossed  at  exactly  right  angles  the  plate  produces  only  a 
slight  brightening. 

Since  rotatory  power  manifests  itself  only  in  the  optic-axial 
direction  and  in  directions  immediately  adjacent  to  the  same,  then, 
viewed  in  convergent  light,  a  plate  of  an  optically  active  uniaxial 
crystal  cut  perpendicular  to  the  axis  must  likewise  exhibit,  when 
the  nicols  of  the  polariscope  are  crossed,  the  black  cross  with  the 
circular  color  rings, — but  with  this  exception:  that  part  of  the 


230  ROTATION    OF   THE   POLARIZATION    PLANE 

field  at  which  the  rays  traversing  the  crystal  parallel  to  the  axis 
converge,  i.e.  the  center  and  its  immediate  vicinity,  cannot 
appear  dark,  because  it  is  of  precisely  these  rays  that  the  polar- 
ization plane  is  rotated.  The  center  of  the  rings  must  there- 
fore exhibit  the  same  color  *  that  the  whole  plate  exhibits  in 
parallel  light;  and  accordingly,  with  different  plate  thickness, 
a  different  color.  So  too  must  this  color  change  when  the  ana- 
lyzer is  rotated;  and  from  the  sequence  of  the  arising  colors 
and  the  sense  of  the  rotation  we  shall  be  able,  just  as  in  par- 
allel light,  to  determine  whether  the  plate  is  dextro-  or  levo- 
rotatory. 

If  a  dextro-  and  a  levo-quartz  plate  are  laid  one  upon  the 


Fig.  in. 

other  and  brought  into  the  conoscope  between  crossed  nicols, 
the  center  of  the  interference-figure  must  remain  dark,  because 
the  rotations  of  the  vibration  directions  by  the  two  plates, 
respectively,  exactly  neutralize  each  other.  From  the  dark 
center  of  the  arising  interference-figure  there  proceed,  however, 
no  black  cross  arms,  but  spirally  wound  curves,  — called,  after 
their  discoverer,  Airy's  spirals,  —  the  direction  of  whose  winding 
indicates,  at  the  same  time,  into  which  of  the  two  oppositely 
rotatory  crystals  the  light  rays  first  enter.  Figure  ma  repre- 
sents the  phenomenon  in  the  case  where  the  light  passes  first 

*  Figure  2  of  Plate  II  represents  the  interference-figure  of  a  quartz  plate  cut 
perpendicular  to  the  axis. 


ROTATION   OF   THE    POLARIZATION  PLANE  231 

through  the  levo-plate,  Fig.  nib  that  where  it  passes  first 
through  the  dextro-plate.*  Airy's  spirals  are  likewise  obtained 
if  instead  of  the  second  quartz  plate  one  employs  a  circular  an- 
alyzer; i.e.  the  combination  of  a  nicol  with  a  quarter-undulation 
mica  plate. 

Now  all  that  has  been  said  applies  also  to  the  remaining 
substances  whose  crystals  are  optically  active,  but  with  this 
difference:  their  rotatory  power  is  not  the  same  as  that  of 
quartz,  being  with  some  higher  but  with  the  greater  number 
lower. 

In  the  optically  biaxial  crystals  there  is  no  direction  without 
double  refraction;  for,  although  the  rays  parallel  to  the  so-called 
optic  axes  have  a  common  ray-front,  yet  there  correspond  to 
them  an  infinity  of  vibrations,  with  all  possible  azimuths.  To 
demonstrate  the  occurrence  of  polarization-plane  rotation,  there- 
fore, is  here  a  matter  surrounded  by  difficulties.  Nevertheless, 
in  a  plate  of  cane  sugar  one  centimeter  thick  Pocklington  has 
recently  been  able  to  prove  that  rotation  takes  place  at  the  center 
of  the  dark  brush  of  each  of  the  two  axes;  at  the  one  center 
a  levo  rotation  o'f  22°,  at  the  other  a  dextro  rotation  of  64°. 
Again,  Dufet,  still  more  recently,!  has  shown  that  rotatory 
power  is  possessed  by  a  more  extensive  series  of  biaxial  crystals, 
and  especially  that  it  is  of  equal  magnitude  parallel  to  the  two 
optic  axes  of  a  crystal  if  the  axial  angle  is  bisected  by  a  plane 
of  optical  symmetry,  else  different.  (As  is  seen  from  the  ex- 
ample of  cane  sugar,  the  sense  of  the  rotation  can  even  be  the 
opposite.)  Moreover,  it  was  shown  earlier  that  quartz  that  by 
pressure  has  become  biaxial  still  possesses  rotatory  power.  (See 
p.  277,  under  "Influence  of  Other  Properties  on  the  Optical 
Properties".) 

*  Among  the  crystals  of  quartz  occurring  in  nature  are  such  as  are  composed 
of  dextro-  and  of  levo-quartz  in  layers,  and  this  composition  is  then  often  to  be 
recognized  only  by  the  appearance  of  Airy's  spirals. 

f  In  an  article  ("  Recherches  experimentales  sur  1'existence  de  la  polarisation 
rotatoire  dans  les  cristaux  biaxes."  Bull.  soc.  franc,  de  mineral.  1904,  27,  156 
et  seq.)  that  appeared  while  this  work  was  in  press. 


232  ABSORPTION  OF  LIGHT   IN   CRYSTALS 

ABSORPTION  OF  LIGHT  IN  CRYSTALS 

On  its  way  through  every  body  light  suffers  a  weakening  of 
its  intensity,  a  partial  absorption;  when  this  is  very  slight,  we 
say  the  body  is  transparent;  when  it  is  so  pronounced  that  even 
after  traveling  only  a  short  distance  the  light-motion  is  anni- 
hilated, we  call  the  body  opaque.*  Secondly,  this  absorption 
lays  hold  of  the  differently  colored  light  rays  in  a  different  de- 
gree; so  that  the  light  emerging  from  the  body  not  only  has 
no  longer  its  former  brightness,  but  also  no  longer  its  former 
color.  If  the  diversity  in  the  strength  of  the  absorption  for 
the  different  wave  lengths  in  the  light  is  but  slight,  the  dif- 
ferent wave  lengths  will  combine  to  form  a  color  impression 
so  little  different  from  that  of  the  entering  light  that  we  are 
unable  to  distinguish  the  one  from  the  other.  We  then  speak  of 
the  body  as  colorless.|  But  an  absolutely  colorless  substance 
exists  just  as  little  as  one  that  is  absolutely  transparent.  And 
even  in  bodies  that  to  us  seem  altogether  colorless  the  different 
colors  composing  the  white  light  are  not  all  absorbed  in  exactly 
the  same  degree;  so  that,  if  we  but  cause  the  white  light  to  pass 
over  a  very  long  distance  through  the  crystal,  the  rays  that  are 
the  more  strongly  absorbed  fall  perceptibly  back  in  intensity, 
relatively  to  the  others,  and  the  emerging  rays  then  no  longer 
produce  the  impression  of  white,  but  of  a  color.  Very  many 
substances,  however,  absorb  the  different  kinds  of  light  so 
differently  that  even  in  thin  layers  of  them  the  white  light  be- 
comes vividly  colored;  these  it  is,  that  are  designated  especially 
as  colored  substances.  Many  absorb  light  only  of  one  or  of 
several  definite  wave  lengths,  this  absorption  being  complete  if 

*  Hence  it  follows  that  the  two  words  express  only  relative  ideas,  and  that 
neither  an  absolutely  transparent  nor  an  absolutely  opaque  body  exists. 

f  Such  bodies  can,  however,  exert  a  very  strong  absorption  upon  ether  vibra- 
tions whose  period  is  greater  or  less  than  those  of  the  visible  spectrum;  i.e.  on 
infra-red  or  ultra-violet  vibrations. 


ABSORPTION   OF   LIGHT   IN   CRYSTALS  233 

the  layer  is  of  some  thickness;  so  that  the  transmitted  light,  de- 
composed by  a  prism  of  high  dispersive  power,  yields  a  spectrum 
that  is  broken  by  dark  stripes  (absorption  spectrum).  Other 
substances  absorb  the  majority  of  colors  far  more  strongly  than 
they  do  one  or  several  of  them;  so  that  these  latter  alone  appear 
in  the  spectrum,  as  bright  stripes  or  bands.  But  there  is  no 
body  that  would  transmit  the  light  only  of  one  definite  wave 
length  and  absorb  all  the  rest  in  equal  degree, —  no  body,  there- 
fore, that  might  serve  to  produce  really  monochromatic  light. 
With  many  red  glasses,  to  be  sure,  the  .  absorption  of  all  the 
colors  except  red  is  as  good  as  complete;  yet,  that  the  trans- 
mitted light  consists  of  various,  although  not  very  many,  wave 
lengths  follows  from  the  spectral  decomposition,  this  always 
yielding  a  broad  red  band.  Absolutely  homogeneous  light  can- 
not be  obtained  through  absorption,  but  only  through  emission 
from  glowing  bodies;  e.g.  from  the  incandescent  metallic  vapors 
already  mentioned  for  that  purpose. 

The  coloring  of  a  crystal  can  be  either  allochromatic,  i.e. 
not  belonging  to  its  own  substance,  but  arising  from  the  admix- 
ture of  foreign  matter;  or  idiochromatic,  i.e.  an  essential  attri- 
bute of  its  material  nature.  Among  the  latter  colorings  there 
belong  the  yellow  and  red  colors  of  the  chromic  acid  salts  and 
of  many  organic  nitro-  and  azo-compounds,  also  the  blue  and 
green  colors  of  the  copper  salts.  When  the  coloring  of  a  crystal 
is  allochromatic,  the  coloring  body  may  be  disseminated  in  small 
isolated  particles,  which  can  be  recognized  in  the  thin  section  of 
the  enclosing  crystal  by  means  of  the  microscope  (if  these  par- 
ticles are  regularly  distributed,  then  in  reflected  light  they  give 
rise  to  a  so-called  "schiller",  a  bronze-like  luster,  in  definite 
directions);  or  else  the  coloring  body  exists  in  the  crystal  in  a 
dissolved  (dilute)  state,  so  to  speak,  this  being  true  especially  as 
regards  many  organic  pigments.  In  the  case  of  dilute  coloring 
the  crystal  is  apparently  homogeneous,  like  the  solution  of  a 
pigment  in  a  liquid,  wherefore  it  may  then  be  designated  as  a 
"  solid  solution";  such  a  crystal,  in  those  of  its  optical  proper- 


234  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

ties  that  depend  on  the  color,  behaves  like  a  crystal  whose  color- 
ing is  idiochromatic. 

*"~  The  so-called  body  color  *  of  the  homogeneously  colored  crys- 
tals, i.e.  the  color  these  crystals  impart  to  the  white  light  rays 
passing  through  them,  is  only  then  the  same  in  all  directions, 
when  the  vibrations  of  all  directions  are  transmitted  in  the  same 
manner  and  hence  equally  fast;  but,  of  all  crystals,  this  is  the 
case  only  with  the  singly  refracting.  With  the  crystals  of  all  the 
four  other  groups  (see  p.  196),  i.e.  with  the  optically  uniaxial 
and  biaxial,  the  transmission  of  light  in  different  directions  is 
different,  and,  accordingly,  so  too  are  the  kind  and  the  magni- 
tude of  the  absorption:  with  these  crystals  the  light  rays  that 
have  passed  through  an  equal  thickness  but  in  a  different  direc- 
tion possess  different  brightness  and  different  color.  The  latter 
property,  to  exhibit  a  different  color  in  different  directions,  — 
which  property  therefore  can  belong  only  to  the  doubly  refract- 
ing crystals,  —  is  termed  pleochroism;  of  this  property  the  laws 
have  been  investigated  chiefly  by  Brewster  and  Haidinger. 

Like  its  transmission,  the  absorption  also  of  light  is  an 
ellipsoidal  property  of  crystals;  for  the  most  proper  treatment 
of  it  we  take  into  account  a  quantity  called  the  coefficient  of 
absorption  and  denoted  by  a.  This  quantity  is  specific  to  the 
absorbing  substance,  and  its  value,  besides  depending  therefore 
on  the  nature  of  the  substance,  is  further  conditioned  by  the 
direction  of  the  vibrations  and  by  the  color  of  the  light.  If  we 
denote  the  intensity  of  the  incident  rays  by  I0,  that  of  the  rays 
emerging  from  the  absorbing  body  by  /,  and  the  thickness  of  the 
latter  by  z;  and  if  e  be  the  base  of  the  natural,  or  Naperian, 
logarithms,  then  the  following  law  applies: 


According  to  this  equation,  therefore,  a  may  be  calculated 
from  the  loss  in  brightness  suffered  by  rays  of  known  intensity 

*  [Often  designated  simply  as  the  color  in  transmitted  light,  while  the  "  sur- 
face color"  (see  p.  249)  is  known  also  as  the  color  in  reflected  light.] 


ABSORPTION    OF   LIGHT   IN   CRYSTALS  235 

in  their  passage  through  a  crystal  whose  thickness  is  measured 
in  the  direction  in  question.  Hence,  if  from  a  point  within  the 
crystal  one  imagines  as  laid  off  in  every  direction  a  length  pro- 
portional to  — = ,  this  length  may  be  said  to  be  a  measure  of  the 

remaining  brightness  of  a  plane-polarized  ray  vibrating  along 
the  direction  in  question.  If  we  connect  the  extremities  of  all 
these  lengths  with  one  another,  we  obtain  a  closed  curved  sur- 
face (in  general  a  triaxial  ellipsoid)  which  gives  us,  for  a  definite 
color,  a  mental  picture  of  the  brightness  of  the  light  rays  vibrat- 
ing in  all  possible  crystallographic  directions,  just  as.  does  the 
index-surface  a  mental  picture  of  the  transmission  of  those  rays. 
This  surface,  designated  by  Mallard  as  the  "  ellipsoide  inverse 
d'absorption",  we  shall  call  briefly  the  absorption-surface.  Like 
the  index-surface,  it  has  for  a  different  color  a  different  form, 
perhaps  also  a  different  crystallographic  orientation. 

a.  SINGLY  REFRACTING  CRYSTALS.  — The  absorption-surface 
has  for  every  color  the  form  of  a  sphere.     Vibrations  of  a  defi- 
nite color,  whatever  their  crystallographic  orientation,  suffer  the 
same  absorption;   for  a  different  color,  however,  the  absorption- 
surface  has  a  different  diameter.     In  case  the  values  of  the 
coefficient  of  absorption   for   different  colors  are  very  unlike, 
the  white  light,  even  after  it  has  passed  through  only  a  slight 
thickness  of  the  crystal,  will  be  vividly  colored;  but  in  all  direc- 
tions, when  the  light  has  traversed  an  equal  thickness  of  the 
crystal,  the  color  is  exactly  the  same. 

b.  OPTICALLY  UNIAXIAL  CRYSTALS. — The  absorption-surface 
is  a  rotation  ellipsoid,  whose  rotation  axis  is  the  optic  axis  of  the 
crystal;    in  other  words,  only  those  rays  undergo  equal  absorp- 
tion whose  vibration  directions  form  equal  angles  with  the  optic 
axis,  —  only  those  rays,  therefore,  that  in  the  crystal  are  trans- 
mitted with  equal  velocity.     Hence,  for  the  light  of  a  definite 
color  the  absorption  is  greatest  for  vibrations  parallel  to  the  axis, 
diminishes  with  increasing  inclination  to  the  axis,  and  is  least  for 
all  rays  vibrating  perpendicular  to  the  axis;  or  else  the  reverse 


236  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

takes  place,  the  minimum  absorption  being  suffered  by  the  rays 
that  vibrate  parallel  to  the  axis  and  the  maximum  by  those  / 
vibrating  perpendicular  to  it.  In  general  the  rule  applies  that 
was  laid  down  by  Babinet;  namely,  that  the  uniaxial  crystals 
\\ith  positive  double  refraction  present  the  former  of  the  two 
cases  distinguished,  absorbing  the  extraordinary  ray  the  more 
strongly,  while  in  negative  crystals  it  is  the  vibrations  perpen- 
dicular to  the  axis,  the  ordinary  rays  therefore,  that  are  weakened 
the  most.  To  express  this  rule  more  briefly:  the  ray  the  more 
strongly  refracted  is  also  the  more  strongly  absorbed. 

For  different  colors  the  absorption-surface  has  not  only  a 
different  size  but  also  a  different  form;  in  other  words,  the 
coefficient  of  absorption  is  not  only  different  in  an  absolute 
sense,  but  its  value  for  vibrations  parallel  to  the  axis  stands  in 
a  different  ratio  to  its  value  for  vibrations  perpendicular  to  the 
axis.  In  consequence  of  this,  while  all  rays  whose  vibration 
directions  form  the  same  angle  with  the  optic  axis  will  indeed 
(as  in  the  singly  refracting  crystals)  experience  equal  absorption 
for  each  individual  color,  yet  with  a  different  common  inclination 
of  the  vibration  directions  to  the  axis  the  relative  brightness  of 
these  colors  will  be  different;  therefore  the  arising  composite 
color  will  be  the  same  only  for  rays  falling  at  equal  inclination 
to  the  axis:  with  increasing  or  diminishing  inclination,  on  the 
other  hand,  the  color  will  vary,  varying  in  the  opposite  sense 
when  the  vibration  direction  of  the  rays  approaches  and  when 
it  recedes  from  the  optic  axis.  Let  us  denote  by  0  the  body 
color  of  the  crystal  that  arises  with  a  definite  thickness  of 
the  crystal  when  white  light  enters  it  parallel  to  the  optic  axis. 
Of  these  rays  the  vibration  directions,  although  in  all  possible 
azimuths,  are  nevertheless  always  perpendicular  to  the  axis;  and 
these  vibration  directions  are  all  equivalent,  corresponding  there- 
fore to  the  same  absorption.  A  light  ray  passing  through  an 
equal  thickness  of  the  same  crystal  but  perpendicular  to  the 
axis  is  split  up  into  two  rays,  of  which  one  vibrates  perpendicu- 
lar, the  other  parallel,  to  the  axis.  The  former  ray,  whatever 


ABSORPTION   OF   LIGHT   IN   CRYSTALS  237 

its  direction  otherwise  may  be,  exhibits  the  absorption-color  0; 
the  latter  a  different  color,  which  we  will  call  e,  and  which 
according  to  the  above  is  obviously,  of  all  the  absorption-colors 
of  this  crystal,  the  one  differing  the  most  from  0.  If  we  look  at 
the  crystal  in  such  a  way  that  the  light  falls  through  it  in  the 
direction  perpendicular  to  the  axis,  then  both  of  the  polarized 
rays  —  that  with  the  color  0  and  that  with  the  color  e  —  reach  our 
eye  simultaneously;  and  we  are  unable  to  separate  them  from  each 
other,  but  receive  the  resultant  impression  — of  a  color  that  we 
shall  denote  by  0  +  0.  Now  the  colors  0  and  0  +  z,  i.e.  the  body 
colors  of  the  crystal  parallel  and  perpendicular  to  the  axis,  are 
obviously  the  more  different,  the  more  0  and  z  themselves  differ 
from  each  other.  Hence,  in  the  intermediate  directions  the 
body  color  of  the  crystal  is  intermediate,  being  the  more  like  0, 
the  more  nearly  the  direction  in  which  the  light  falls  through 
the  crystal  approximates  to  the  optic  axis,  and  vice  versa.* 
When  the  color  0  differs  but  little  from  e,  then  from  the  com- 
posite color  0  +  e  it  differs  still  less;  and  in  such  cases  as  this 
the  crystal,  if  observed  without  further  aid,  appears  to  have 
the  same  body  color  in  all  directions.  More  especially  is  this 
true  of  the  so-called  colorless  substances;  because,  with  them, 
even  the  sum  total  of  the  absorption  is  imperceptible,  its  dif- 
ference in  different  directions  accordingly  escaping  all  obser- 
vation. Therefore  strong  pleochroism,  i.e.  great  variation  of 
the  body  color  with  the  direction,  can  be  exhibited  only  by 
crystals  having  high  "absorbency " ;  i.e.  crystals  that  absorb 
the  light  very  strongly,  hence  being  vividly  colored.  Even 
among  these  there  are  many  that  have  only  a  slight  degree  of 
pleochroism,  the  body  colors  they  exhibit  parallel  and  perpen- 
dicular to  the  axis  thus  being  very  similar. 

In  the  cases  last  mentioned,  in  order  to  perceive  the  exist-- 
ence  of  pleochroism — whereby,  at  the  same  time,  that  of 

*  Therefore  it  is  not  entirely  correct  to  call  this  phenomenon  —  as  it  fre- 
quently is  called  —  by  the  name  "dichroism";  because  it  is  not  a  matter  of  two 
colors,  but  of  a  continuous  series  of  colors  between  two  extremes. 


238  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

double  refraction  is  verified  —  service  must  be  made  of  a  little 
apparatus  constructed  by  Haidinger,  known  as  the  dichroscope 

or  dichroscopic  lens.     This 
consists  of  a  rhombohedron 
° K     of    calcite,    whose    section, 
abed,  is  represented  in  Fig. 


Fig.  112.  II2    ust  as    n      8-  37> 

81;  so   that  ab  and  cd  are 

the  short  diagonals  of  two  opposite  rhombohedron  faces.  The 
rhomb  is  mounted  in  a  brass  tube,  and  both  before  and  behind 
it,  touching  the  calcite,  there  is  a  glass  wedge  (g  and  g')  so  fitted 
that  the  faces  of  entrance  and  emergence  of  the  light  stand 
perpendicular  to  the  rhombohedron  edges  ac  and  bd]  at  these 
faces,  therefore,  there  is  no  refraction  of  the  rays  that  pass 
through  the  instrument  parallel  to  these  edges.  In  front  the 
mounting  has  a  wide,  round  opening  for  looking  inside,  the  eye 
being  then  at  £;  behind,  on  the  other  hand,  there  is  a  square 
opening,  o,  of  only  2  or  3  mm.  diameter,  through  which  the  light 
falls  when  this  end  is  directed  toward  the  bright  sky  or  some 
other  source  of  light.  Immediately  in  front  of  the  glass  wedge 
g  is  a  plano-convex  lens,  /,  by  means  of  which,  if  the  calcite 
were  not  present,  the  eye  at  E  would  see  at  the  distance  of 
distinct  vision  a  magnified  virtual  image  of  the  bright  opening  o. 
By  reason,  however,  of  the  double  refraction  in  the  calcite,  two 
such  images  appear,  and  one  of  these,  since  the  extraordinary 
is  deflected  in  the  principal  section,  appears  exactly  above  the 
other.  Now  the  length  of  the  rhombohedron  is  so  chosen  that 
the  two  images  do  not  partially  coincide,  but  that  the  upper 
edge  of  the  lower  image  just  touches  the  lower  edge  of  the 
upper. 

Hence,  if  a  uniaxial  crystal  is  held  before  the  small  opening  * 

*  V.  von  Lang  suggests  that  it  is  well  to  cover  the  end  K  of  the  tube  (Fig. 
112)  with  a  rotatable  metal  cap  having  in  it  a  circular  opening;  over  this  opening 
the  crystal  is  fastened  with  wax  and  hence,  by  rotating  the  cap,  can  easily  be 
brought  into  its  correct  position. 


ABSORPTION    OF    LIGHT   IN   CRYSTALS 


239 


in  such  a  way  that  the  rays  passing  through  parallel  to  the  long 
axis  of  the  dichroscope  first  traverse  the  crystal,  in  a  direction 
perpendicular  to  its  optic  axis,  and  if  in  addition  the  crystal  is  so 
held  that  its  axis  is  at  the  same  time  parallel  to  the  principal 
section  of  the  calcite,  then  the  two  rays  arising  in  the  crystal 
enter  the  calcite  in  such  wise  that  the  vibration  direction  of 
the  ordinary  ray  is  parallel  to  that  of  the  ordinary  ray  in  the 
calcite,  and  the  vibrations  of  the  extraordinary  parallel  to 
those  of  the  extraordinary  in  the  calcite.  So  neither  of  the 
two  rays  suffers  a  fresh  splitting- up  in  the  calcite;  and  thus, 
of  the  two  images  of  the  bright  opening 
in  the  dichroscope,  the  one  is  formed  only 
by  the  rays  that  on  emerging  from  the  crys- 
tal to  be  investigated  were  vibrating  par- 
allel to  its  axis,  the  second  image  only  by 
those  rays  whose  vibration  direction  in  the 
same  crystal  was  perpendicular  to  the  axis.  "* 
The  color  of  the  latter  image,  therefore, 
must  be  the  one  previously  denoted  by  o; 
the  color  of  the  former,  that  denoted  by  e. 
This  is  readily  seen  from  Fig.  113,  where 
klk2k3k4  is  the  outline  of  the  crystal  to  be 
investigated,  A  A'  its  optic-axial  direction, 
and  consequently  oor  the  vibration  direction  of  the  ordinary  ray 
emerging  from  it,  ee'  that  of  the  extraordinary;  abed  is  the  cross- 
section  of  the  calcite  rhombohedron,  whose  principal  section  is 
parallel  to  that  of  the  crystal;  OHD'  is  the  vibration  direction  of 
the  light  in  the  one  image  of  the  square  opening  covered  by  the 
crystal,  ee'  the  vibration  direction  in  the  other  image.  Hence, 
if  one  rotates  the  crystal  about  the  axis  of  the  dichroscope  as  an 
axis,  oo'  and  ee'  form  an  angle  with  u)a>f  and  ee',  wherefore  in 
the  calcite  each  of  the  two  rays  is  doubly  refracted,  and  con- 
tributes to  each  of  the  two  images;  when  the  angle  mentioned 
amounts  to  45°  the  component  contributed  by  each  ray  to  each 
image  is  half  its  brightness,  and  so  the  two  images  are  of  the 


240  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

same  color  —  of  that  color,  0  +  e,  that  the  crystal  exhibits  in 
the  direction  in  question  to  the  unassisted  eye.  After  90°  rota- 
tion the  two  images  of  the  dichroscope  again  exhibit  the  greatest 
difference  in  coloring,  but  their  colors  have  exchanged  places. 
And  so  on. 

Always,  therefore,  when  the  optic  axis  of  the  crystal  to  be 
investigated  for  pleochroism  is  parallel  to  the  vibration  direction 
of  either  of  the  two  rays  transmitted  in  the  calcite,  the  one 
image  exhibits  the  color  0,  the  other  the  color  e.  Since,  then, 
these  colors  differ  from  each  other  more  than  do  the  colors 
o  and  o  +  *,  visible  in  the  crystal  parallel  and  perpendicular  to 
the  axis,  and  since,  besides,  in  this  instrument  the  two  color- 
ings are  seen  simultaneously  and  in  contact,  side  by  side,  — • 
under  which  circumstances  even  very  slight  differences  between 
their  shades  can  be  perceived, —  it  is  clear  that  with  the  aid  of 
this  simple  apparatus  one  can  ascertain  the  existence  of  dichro- 
ism  in  a  crystal  even  when  it  is  rather  weak,  while  in  order  to 
recognize  it  without  the  dichroscopic  lens  there  must  be  a  very 
considerable  difference  in  the  absorption,  such  as  is  exhibited  by 
only  a  limited  number  of  substances. 

It  is  easy  to  see  that,  when  the  crystal  is  so  held  before  the 
instrument  that  the  rays  traverse  it  oblique  to  its  optic  axis, 
the  color  of  the  one  image  will  be  a  and  the  color  of  the  other  a 
tint  lying  between  0  and  e;  finally,  that  when  the  light  passes 
through  the  crystal  parallel  to  the  axis,  however  the  crystal  may 
be  rotated  in  its  own  plane,  both  images  must  exhibit  the  same 
color,  0. 

As  examples  of  very  highly  pleochroic  uniaxial  crystals,  two 
minerals  may  be  mentioned:  First,  chlorite  (penninite),  whose 
color  as  seen  through  a  plate  perpendicular  to  the  axis  is 
emerald-green,  and  as  seen  through  a  plate  parallel  to  the  axis 
brownish  red;  second,  tourmaline,  which  occurs  in  very  diverse 
colors,  many  varieties  of  the  mineral  presenting  at  the  same 
time  an  example  of  dilute  coloring  by  foreign  pigments.  In  the 
latter  crystals,  as  in  those  that  are  colored  by  their  iron  content 


ABSORPTION    OF   LIGHT   IN   CRYSTALS  241 

and  therefore  idiochromatically,  the  ordinary  ray  is  so  strongly 
absorbed  that  tourmaline  plates  cut  parallel  to  the  axis  transmit 
almost  no  light  except  the  extraordinary.  (Cf.  p.  54.) 

In  the  case  of  certain  crystals  having  strong  pleochroism,  by 
means  of  a  plate  cut  perpendicular  to  the  axis  the  difference  in 
the  absorption  of  rays  passing  through  the  plate  parallel  and 
inclined  to  the  axis  may  be  perceived  directly,  without  instru- 
mental aid.  For  example,  if  one  holds  near  the  eye  a  plate  of 
magnesium  platino-cyanide  cut  or  split  in  the  direction  stated 
(the  crystals  of  this  salt  cleave  perfectly  along  a  plane  perpen- 
dicular to  the  axis),  and  looks  through  it  at  a  white  surface 
(best  a  uniformly  white  bank  of  clouds) ,  one  sees  a  circular  violet 
spot  upon  a  background  of  vermilion.  This  phenomenon  is 
explained  as  follows:  The  magnesium  platino-cyanide  trans- 
mits blue  rays  only  along  the  axis  and  at  very  slight  inclinations 
to  the  axis,  —  and  even  here,  only  when  the  plate  is  very  thin, — 
while  in  other  directions  it  transmits  only  red  rays;  in  conse- 
quence the  light  passing  through  along  the  axis  becomes  violet- 
colored,  but  with  a  certain  inclination  to  this  direction  the  blue 
is  absorbed,  both  on  account  of  the  increasing  thickness  of 
crystal  traversed  and  because  of  the  deviation  from  the  axis.  If 
a  nicol  is  held  before  or  behind  the  plate,  then,  of  the  rays 
inclined  to  the  axis  in  the  vibration  plane  of  the  nicol,  the  ordi- 
nary —  i.e.  violet-colored  — portion  is  extinguished,  while  of  the 
rays  inclined  in  a  plane  perpendicular  to  that  plane  the  same 
portion  is  transmitted;  parallel  to  the  vibration  direction  of 
the  nicol,  upon  the  violet  background,  there  then  appear  two 
red  brushes  of  a  shape  similar  to  that  portrayed  in  Fig.  114, 
page  246. 

c.  OPTICALLY  BIAXIAL  CRYSTALS. —  Here  the  absorption-sur- 
face for  a  definite  color  is  in  general  a  triaxial  ellipsoid,  the  lengths 
of  whose  three  axes,  the  so-called  absorption-axes  (i.e.  the  vibra- 
tion directions  of  the  greatest,  the  intermediate,  and  the  least 
absorption),  stand  in  ratios  that  are  entirely  independent  of  the 
ratios  holding  good  with  the  index-surface;  in  direction,  too,  as 


242  ABSORPTION    OF  LIGHT  IN   CRYSTALS 

a  general  thing,  these  axes  do  not  coincide  with  those  of  the 
index-surface.  For  a  different  color  the  absorption-surface  has 
not  only  a  different  size  but  also  a  different  form,  —  in  other 
words,  different  ratios  among  its  axes. 

When,  however,  the  crystal  belongs  to  the  group  of  biaxial 
crystals  having  the  highest  symmetry,  i.e.  when  the  three  axes 
of  the  index-surface  have  the  same  crystallographic  orientation 
for  all  colors  (cf.  p.  168),  then  do  likewise  the  three  absorption- 
axes  for  all  colors  coincide;  they  coincide  with  one  another  and 
also  with  the  three  mutually  perpendicular  axes  of  the  index- 
surface.  In  this  case  not  only  the  transmission,  but  also  the 
absorption,  of  light  is  absolutely  symmetrical  to  the  three  prin- 
cipal optic  sections  of  the  crystal.  Let  us  consider  this  case 
first,  it  being  the  most  simple.  If  a  denote  the  color  of  the  rays 
that  have  as  their  vibration  direction  in  the  crystal  the  X-axis  of 
the  index-surface,  fc  the  color  of  those  vibrating  parallel  to  the 
F-axis,  and  t  that  of  the  rays  vibrating  parallel  to  the  Z-axis, 
then  in  the  dichroscope,  on  our  looking  through  a  plate  cut  per- 
pendicular to  the  F-axis,  when  X  and  Z  are  parallel  to  the 
vibration  directions  of  the  calcite  we  shall  see  in  the  one  image 
the  color  a,  in  the  other  image  the  color  c.  But  if  we  take  a  plate 
cut  normal  to  Z,  then,  with  the  plate  in  the  analogous  position 
relative  to  the  instrument,  the  one  image  exhibits  a,  the  other  fr. 
Finally,  a  plate  cut  perpendicular  to  X  exhibits  the  colorings  b 
and  t  separately,  if  it  is  so  held  before  the  opening  of  the  dichro- 
scopic  lens  that  either  F  or  Z  is  parallel  to  the  principal  section 
of  the  calcite  contained  in  the  instrument  Therefore,  when 
the  dichroscope  is  employed  we  need  observe  the  crystal 
only  in  two  of  the  directions  named,  in  order  to  determine  the 
three  so-called  axial  colors  —  the  colors  arising  by  the  absorp- 
tion of  white  light  rays  vibrating  parallel  to  the  three  absorption- 
axes. 

Without  the  dichroscope,  on  the  other  hand,  we  are  able  to 
see  none  of  these  colors  separately.  For,  when  we  look  through 
a  plate  cut  from  the  crystal  normal  to  X,  for  example,  our  eye 


ABSORPTION   OF  LIGHT  IN   CRYSTALS  243 

receives  simultaneously  the  rays  vibrating  parallel  to  F,  with  the 
color  b,  and  those  vibrating  parallel  to  Z,  with  the  color  c,  so  that 
we  shall  see  a  coloring  fc  +  c,  composed  of  both  these  colors;  while 
in  just  the  same  way  a  plate  cut  normal  to  F  shows  us  a  com- 
posite color  a  +  c,  and  a  plate  whose  faces  stand  perpendicular 
to  the  Z-axis  a  +  fc.*  Hence,  it  is  clear  that  the  color  impressions 
made  up  of  two  axial  colors,  viz.  a  +  ft,  a  +  c,  and  b  +  c,  will 
be  less  different  from  one  another  than  are  the  axial  colors 
themselves.  Therefore,  for  the  same  reasons  as  were  adduced 
on  page  240  in  discussing  the  uniaxial  crystals,  one  can  recog- 
nize with  the  dichroscope  a  far  lower  degree  of  pleochroism  than 
without  this  instrument.  With  it,  when  there  is  high  pleochroism, 
one  can  very  easily  distinguish  a  uniaxial  crystal  from  a  biaxial; 
because  with  a  crystal  of  the  latter  kind  there  exists  no  direction 
in  which,  however  the  crystal  be  turned,  the  rays  passing  through 
it  yield  two  images  colored  exactly  alike,  — as  is  the  case  with 
the  rays  parallel  to  the  optic  axis  in  crystals  of  the  former 
kind. 

As  for  the  body  colors  of  such  biaxial  crystals  in  other  direc- 
tions than  along  the  three  absorption-axes,  these  colors  vary 
with  the  direction,  and  their  variation  is  entirely  symmetrical 
to  the  three  principal  optic  sections.  When  the  direction  of 
the  white  light  rays  passing  through  the  crystal  lies  within  one 
of  these  principal  sections,  .the  color  exhibited  is  a  mixture  of 
the  following  two :  First,  a  shade  lying  between  two  of  the  axial 
colors,  namely,  between  the  two  whose  vibrations  take  place 
parallel  to  the  principal  section  in  question;  and  second,  the 
third  axial  color,  —  i.e.  the  color  of  the  ordinary  rays  vibrating 
perpendicular  to  that  principal  section.  In  a  direction  that  falls 
in  none  of  the  three  principal  sections  the  crystal  exhibits  an 

*  This  mixed  color,  which  a  crystal  plate  exhibits  without  the  dichroscope, 
was  called  by  Haidinger  its  facial  color  (Flachenfarbe).  By  the  instrument 
named,  therefore,  the  facial  color  of  a  plate  cut  parallel  to  a  principal  optic  section 
is  resolved  into  the  axial  colors;  stated  generally:  the  facial  color  of  a  plate  hav- 
ing any  chosen  orientation  is  resolved  into  those  two  colors  that  correspond  to  the 
vibration  directions  of  the  plate. 


244  ABSORPTION    OF    LIGHT    IN    CRYSTALS 

absorption-color  lying  between  those  two  of  the  three  axial 
colors  that  present  the  greatest  difference;  there  accordingly 
exist  in  the  crystal  all  possible  color  shades  between  the  two 
that  are  the  most  different;  and  therefore  the  word  'trichroism' 
—  not  infrequently  employed  — is  just  as  little  correct  a  name 
for  the  color  phenomena  of  biaxial  crystals  as  '  dichroism '  is 
for  those  of  the  uniaxial. 

In  those  optically  biaxial  crystals  in  which  the  principal 
vibration  directions  for  the  different  colors  do  not  coincide,  the 
absorption-axes  too,  for  the  different  colors,  diverge  both  from 
these  vibration  directions  and  from  one  another.  When  a 
principal  vibration  direction  has  identical  orientation  for  all 
colors,  it  is  identical  likewise  with  an  absorption-axis,  and  the 
plane  perpendicular  to  it  is  a  plane  of  symmetry  for  the  absorp- 
tion, also,  of  light.  When  the  crystal  has  no  plane  of  symmetry 
for  the  transmission  of  light  (Group  i,  p.  196),  the  same  ap- 
plies to  the  absorption  as  well.  Thus,  for  example,  if  in  ordinary 
light  falling  on  it  at  right  angles  we  look  at  a  plate  of  a  highly 
absorbent  crystal  of  this  kind  cut  perpendicular  to  the  acute 
bisectrix  for  any  one  color,  it  exhibits  a  definite  absorption-color; 
but  if  we  rotate  it  in  either  direction  so  that  the  direction  of  the 
rays  passing  through  is  approached  first  by  the  one  and  after- 
ward by  the  other  optic  axis,  then  in  the  two  directions  the 
color  in  transmitted  light  does  not  vary  in  the  same  way;  be- 
cause the  absorption  is  not  the  same  symmetrically  on  opposite 
sides  of  the  normal  to  the  plate,  the  directions  that  correspond 
to  the  greatest,  the  least,  and  the  intermediate  absorption  being 
oriented  differently  for  every  color  but  always  one-sidedly  oblique 
to  the  plate. 

Thus,  with  reference  to  the  crystallographic  orientation  of 
the  absorption-axes  the  optically  biaxial  crystals  fall  into  the 
same  three  groups  as  with  reference  to  the  principal  axes  of  the 
index-ellipsoid. 

As  examples  of  optically  biaxial  bodies  having  especially 
high  pleochroism  the  following  may  be  mentioned:  i.  Cordier- 


ABSORPTION   OF    LIGHT    IN    CRYSTALS  245 

ite  (called  also  "  dichroite",  on  account  of  this  property), 
whose  coloring  must  be  attributed  to  a  foreign  body  distributed 
in  dilute  form  and  which  exhibits  the  following  axial  colors:  a, 
light  yellow  to  yellowish  brown  (corresponding  to  the  vibrations 
parallel  to  the  vibration  direction  of  greatest  light  velocity) ;  ft, 
light  blue  (absorption-color  of  the  vibrations  having  the  in- 
termediate light  velocity) ;  and  c,  dark  blue  (vibrations  with  the 
least  velocity,  i.e.  of  the  greatest  refraction,  which  therefore 
are  the  most  strongly  absorbed).  2.  Epidote,  which  in  the 
beautiful  ferruginous  variety  from  the  Sulzbachthal  exhibits 
a,  yellow;  ft,  brown;  c,  green.  3.  Glaucophane,  a  mineral  of 
the  amphibole  group,  which  has  the  axial  colors  a,  light  green- 
ish yellow;  ft,  violet;  c,  ultramarine-blue.  The  two  latter  exam- 
ples refer  to  cases  in  which  the  absorption-axes  and  the  principal 
vibration  directions  for  different  colors  are  dispersed. 

Since  the  kind  and  the  intensity  of  the  color  tones,  besides 
depending  on  the  direction  of  the  vibrations,  are  conditioned 
also  by  the  thickness  of  the  plate,  the  colors  have  in  the  fore- 
going, as  in  the  previous,  examples  been  designated  only  in  a 
quite  general  way.  They  could  be  specified  more  exactly  by 
mentioning  the  number  in  Radde's  International  Color  Scale  * 
with  which  the  color  tone  in  question  most  nearly  agrees;  but 
an  actual  determination  of  the  absorption-color  would  naturally 
require  spectral  decomposition  of  the  light  that  has  passed 
through  the  crystal,  together  with  measurement  of  the  intensity 
of  the  light  in  the  several  parts  of  the  spectrum.  In  both  cases 
it  would  be  necessary  to  determine  and  state  the  thickness  of 
the  crystal  plate  investigated. 

When  a  microscopic  crystal  —  in  the  thin  section  of  a  rock, 

*  Since  colors  in  transmitted  light,  colors  such  as  we  observe  in  pleochroic 
crystals,  cannot  very  easily  be  compared  with  colors  in  reflected  light,  like  those 
exhibited  by  Radde's  scale,  it  would  be  desirable,  for  specifying  the  absorption- 
colors  of  crystals  more  precisely,  to  employ  standard  colors  made  from  differently 
colored  glasses;  these  glasses  would  have  to  be  ground  to  the  form  of  a  sharp 
wedge,  in  order  that  the  intensity  of  the  color  in  question  might  be  determined 
by  stating  the  part  of  the  glass,  —  i.e.  its  thickness. 


246  ABSORPTION   OF    LIGHT    IN   CRYSTALS 

for  example  —  is  to  be  examined  for  pleochroism,  it  is  well  to  em- 
ploy the  method  proposed  by  Tschermak;  namely,  to  polarize  the 
light  entering  the  microscope,  by  means  of  a  nicol :  by  so  moving 
the  rotatable  stage,  with  the  preparation,  that  first  the  one  and 
then  the  other  vibration  direction  of  the  crystal  section  in  ques- 
tion coincides  with  the  principal  section  of  the  polarizer,  the 
observer  can  produce  in  the  crystal,  one  after  the  other,  the 
two  absorption-colors  that  correspond  to  its  vibration  directions. 
In  the  dichroscope,  it  will  be  remembered,  the  same  colors  are 
observed  side  by  side. 

BRUSH  PHENOMENA 

If  one  holds  close  to  the  eye  a  plate  cut  perpendicular  to 
one  of  the  two  optic  axes,  either  of  Brazilian  andalusite  or  of 
the  epidote  mentioned,  and  directs  it 
toward  the  bright  sky,  one  sees  on  a 
colored  background  two  dark  brushes  * 
of  the  form  shown  in  Fig.  114;  toward  the 
center  these  brushes  can  be  recognized  as. 
traces  of  rings.  The  same  phenomenon, 
less  intense,  may  be  observed  also  with 
various  other  pleochroic  minerals;  it  is  par- 
Fl8-  "4-  ticularly  beautiful,  finally,  with  the  so-called 

"  Senarmont's  salt", — i.e.  strontium  nitrate  crystallized  from 
an  extract  of  logwood,  which  by  its  taking  up  the  pigment 
of  this  solution  in  dilute  form  is  colored  red.  If  from  a  crys- 
tal of  the  latter  kind  one  takes,  instead  of  the  plate  cut  per- 
pendicular to  an  axis,  a  plate  cut  normal  to  the  acute  bisectrix, 
one  sees  two  double  brushes;  and  in  the  case  of  yttrium  plat- 
ino-cyanide,  whose  optic  axial  angle  is  very  small,  the  brushes 
are  so  close  together  that  they  form  four  red  sectors,  between 
which  the  background,  here  violet-colored,  appears  in  the  form 
of  a  cross.  The  brushes  always  stand  perpendicular  to  the  optic 
axial  plane,  and  their  centers  correspond  to  the  two  axes.  The 

*  [The  so-called  polarization-brushes  (absorption-tufts,  epoptic  figures).] 


BRUSH   PHENOMENA  247 

phenomenon  may  accordingly  be  employed  for  finding  the  posi- 
tion of  the  latter  without  the  aid  of  a  polariscope.  An  explan- 
ation of  all  these  phenomena  was  given  by  W.  Voigt  in  his 
theory  of  the  absorption  of  light  in  crystals. 

A  phenomenon  similar  to  the  brushes  must  be  exhibited  by 
radiating-fibrous  aggregates  of  crystals,  when  these  are  distinctly 
pleochroic.  If  we  imagine,  for  example,  a  plate  composed  of 
radially  arranged  uniaxial  crystals  that  fulfill  this  condition  and 
whose  long  direction  is  parallel  to  their  optic  axis  (or  if  we 
cause  a  pleochroic  crystal  to  rotate  as  described  on  page  73,  so 
that  it  assumes  in  rapid  succession  the  positions  of  the  different 
crystals  of  such  a  plate),  and  if  we  view  the  mass  through  a 
nicol  or,  what  signifies  the  same,  cause  it  to  be  entered  by  plane- 
polarized  light  and  observe  it  without  the  nicol,  — then  the 
following  must  take  place:  The  light  passing  through  the  crys- 
tals that  lie  with  their  long  direction  in  the  principal  section  of 
the  nicol,  vibrates  parallel  to  their  optic  axis;  while,  therefore, 
these  crystals  appear  in  the  color  of  the  extraordinary  ray,  and 
the  adjacent  crystals,  diverging  but  little  from  them,  in  nearly 
that  same  color,  those  crystals  that  lie  farther  off  in  direction 
split  up  the  incident  light  into  an  ordinary  and  an  extraordinary 
ray  and  therefore  exhibit  a  mixture  of  the  colors  of  these 
rays;  passing  on,  finally,  to  the  crystals  that  stand  perpendic- 
ular to  the  vibration  direction  of  the  entering  light,  the  oscilla- 
tions transmitted  through  these  crystals  take  place  perpendicular 
to  the  axis,  so  that  the  crystals  last  named  appear  in  the  color  of 
the  ordinary  rays.  Therefore,  while  with  the  preparation  be- 
tween crossed  nicols  we  should  see  the  black  cross  as  in  every 
radiating-fibrous  mass,  we  behold  with  one  nicol,  when  the 
pleochroism  of  the  single  crystals  is  sufficiently  strong,  what 
is,  to  be  sure,  likewise  a  cross,  but  a  cross  of  which  the  two 
arms  lying  parallel  to  the  vibration  plane  of  the  polarizer  have 
the  color  of  the  extraordinary  rays  and  the  other  two  arms  the 
color  of  the  ordinary  light-vibrations.  On  this  property  of 
pleochroic  radiating-fibrous  masses  depend  Haidinger's  polari- 


248  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

zation  brushes,  the  production  of  which,  according  to  Helm- 
holtz,  is  explained  as  follows:  In  the  so-called  yellow  spot,  the 
most  important  part  of  the  human  retina,  the  radial  nerve 
fibers,  which  elsewhere  in  the  retina  stand  perpendicular  to  its 
surface,  converge  obliquely  toward  the  center  of  the  yellow 
spot;  like  the  majority  of  organic  fibers,  these  fibers  are 
doubly  refractive,  and  it  must  be  assumed  that  of  the 
light  rays  vibrating  parallel  and  perpendicular  to  their  axis 
they  absorb  the  blue  and  the  yellow  unequally.  In  conse- 
quence, if  one  looks  upon  a  uniformly  illuminated  white  sur- 
face and  at  the  same  time  holds  r,  nicol  before  the  eye  in  order 
that  plane-polarized  light  may  enter,  one  beholds,  passing  out 
from  the  point  on  which  the  eye  is  fixed,  two  brownish  yellow 
brushes;  and  between  these  brushes,  in  the  direction  perpen- 
dicular to  them,  bluish  light  appears.  The  brushes  always  lie 
in  the  polarization  plane  of  the  nicol,  thus  standing  at  right 
angles  to  the  vibration  direction  of  the  polarized  rays,  and 
therefore  when  the  polarizer  is  rotated  the  brushes  likewise 
must  rotate;  consequently  this  phenomenon  may  serve  for 
recognizing  plane-polarized  light  as  such,  directly,  and  for  de- 
termining its  vibration  direction.  The  brushes  are  found  with 
special  facility  if  one  directs  the  Haidinger  dichro- 
scope  toward  a  white  bank  of  clouds:  they  then  ap- 
pear in  the  two  bright,  square  images,  and  in  crossed 
I  position,  as  shown  in  Fig.  115,  because  the  images 

are  polarized  perpendicularly  to  each  other.  (The 
arrows  indicate  the  vibration  directions  of  the  light 
in  the  two  fields.)  It  must  be  remarked  that  not 
all  persons  are  able  to  perceive  the  phenomenon  of  Haidinger' s 
brushes  (it  being  very  weak),  while  certain  individuals  can  see 
the  brushes  even  without  a  nicol,  when  their  eye  is  directed 
toward  the  sky,  owing  to  the  partial  polarization  of  the  light 
reflected  from  the  sky. 


SURFACE   COLORS 


SURFACE  COLORS 


249 


The  majority  of  bodies,  no  matter  whether  colored  or  color- 
less, do  not  alter  the  light  reflected  from  them;  so  that  white 
light  falling  on  a  plane  surface  of  these  bodies  is  also  reflected 
as  white  light.  Metals,  as  is  well  known,  behave  otherwise;  for 
example,  the  image  of  a  white  body  reflected  from  a  polished 
copper  plate  appears  not  white,  but  red.  Between  the  metals 
and  the  common  transparent  bodies  there  stand  in  intermediate 
position,  as  it  were,  those  substances  that  are  transparent  for 
certain  light  rays  and  behave  for  others  similarly  to  the  metals, 
reflecting  the  light  with  metallic  luster;  such  media  have  been 
designated  as  bodies  with  a  surface  color  ("  schiller-color  "). 
Here  belong  a  series  of  double  cyanides  of  platinum  with  other 
metals;  also  a  number  of  salts  of  organic  bases,  especially 
aniline  pigments,  and  others.  To  explain  the  peculiar  behavior 
of  these  bodies  it  must  be  assumed  that  the  light  reflected  from 
them  has  first  penetrated  to  a  certain  depth  and  the  different 
colors .  therewith  experienced  unequal  absorption ;  this  absorp- 
tion would  then  necessarily  stand  in  a  definite  relationship  to 
the  absorption  of  the  transmitted  light.  In  fact,  Haidinger  found 
that  the  surface  color  of  these  media  is  in  a  certain  sense  com- 
plementary to  their  body  color.  Accordingly,  since  in  doubly 
refracting  crystals  the  latter  color  differs  for  differently  directed 
vibrations,  then  in  such  crystals  the  surface  color  also  must,  on 
account  of  the  relationship  mentioned,  vary  with  the  direction  of 
the  vibrations.  It  is  to  be  expected  for  example,  that,  though 
the  surface  color  of  a  uniaxial  crystal  were  the  same  on  all  faces 
lying  parallel  to  the  axis,  it  would  be  different  on  the  faces  that 
are  inclined  to  the  axis,  and  different  again  on  the  plane  per- 
pendicular to  the  axis.  -  These  suppositions  are  confirmed  by  the 
phenomena  observed  in  such  crystals;  thus,  for  example,  in  car- 
mine-red magnesium  platino-cyanide  a  natural  face,  or  an  arti- 
ficial one,  that  lies  parallel  to  the  axis  exhibits  a  green  surface 


250  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

color,*  a  face  perpendicular  to  the  axis  a  violet  one.  If  for  ob- 
serving the  surface  color  one  makes  use  of  the  dichroscope,  then 
in  looking  upon  faces  perpendicular  to  the  axis  the  two  images  are 
seen  to  be  of  the  same  color,  while  in  viewing  faces  parallel  to 
the  axis  the  two  images  are  of  different  color.  It  accordingly 
follows  that  the  light  observed  in  the  surface  color  has  experi- 
enced in  the  superficial  layers  of  the  crystal  a  change  similar  to 
that  experienced  by  the  transmitted  light, — having  been  re- 
solved into  two  rays  polarized  perpendicularly  to  each  other 
(or  at  least  behaving  similarly  to  such  rays)  and  unequally 
absorbed.! 

The  laws  of  metallic  reflection  were  worked  out  theoretically 
by  Cauchy  and  Voigt,  and  specially  for  the  doubly  refracting 
crystals  by  Drude,  who  also  put  the  results  to  experimental 
test.  These  researches  show  that  it  is  possible,  through  in- 
vestigation of  the  light  reflected  from  highly  absorbent  crystals, 
to  calculate  the  refractive  indices  and  absorption  coefficients 
for  differently  directed  vibrations  of  the  light  penetrating  those 
crystals;  but  this  may  be  done  only  when  the  reflecting  surface 
is  absolutely  clean;  and  absolute  cleanness  is  exhibited,  essen- 
tially, only  by  fresh  cleavage-surfaces.  In  this  way  very  high 
refractive  indices,  with  relatively  low  values  for  the  absorbency, 
were  found  in  the  case  of  the  singly  refracting  metals  lead, 
platinum,  and  iron;  the  highest  refringency  was  found  in  the 
case  of  galena  (lead  glance)  and  of  the  optically  biaxial  stibnite 
(antimony  glance);  while  silver  and  gold,  for  example,  have 
very  low  refractive  indices  (0.2  to  0.4,  the  velocity  of  light  in 

*  The  body  color  of  this  salt,  which  in  a  certain  sense  is  complementary  to 
the  surface  color,  is  of  course  pleochroic,  being  for  the  vibrations  parallel  to  the 
axis  a  pure  carmine-red  and  for  the  vibrations  perpendicular  to  the  axis  a  violet. 
This  latter  color  is*-  essentially  identical  with  the  one  exhibited  by  a  plate  cut  per- 
pendicular to  the  axis,  in  the  case  of  perpendicular  reflection;  so  the  reflection 
of  vibrations  taking  place  perpendicular  to  the  axis  is  almost  normal,  the  crys- 
tal behaving,  for  these  vibrations,  almost  like  a  crystal  without  surface  color. 

t  Particulars  as  to  such  phenomena  will  be  found  in  B.  Walter:  "Die  Ober- 
flachen-  oder  Schillerfarben".  Braunschweig,  1895.  (Excerpt  in  Zeitschr.  f.  Kryst. 
28,  632.) 


FLUORESCENCE        .  251 

them  being  greater  than  in  empty  space)  and  a  remarkably 
high  absorbency.  Phenomena  of  double  refraction  of  the 
transmitted  light  were  observed  by  Kundt  in  extremely  thin 
metallic  films  produced  by  deposition  through  electric  dis- 
charge from  cathodes. 

FLUORESCENCE 

When  light  is  absorbed  in  a  body  the  portion  that  appears 
to  be  lost  is  really  converted  into  another  kind  of  motion  — 
into  a  motion  of  the  particles  of  the  body;  that  is,  into  heat. 
In  certain  bodies,  however,  this  latter  kind  of  motion  excites 
vibrations  again  in  the  ether,  these  vibrations  having  a  dif- 
ferent period.  In  such  cases,  under  the  influence  of  an  irradia- 
tion, the  interior  of  the  body  in  its  turn  emits  light,  but  the 
emitted  light  is  of  a  different  color  from  that  absorbed.  This 
property  is  possessed  in  particular  by  the  solutions  of  a  series  of 
organic  substances;  it  appears  likewise  in  fluor-spar  (fluorite), — 
after  which  the  property  is  named, —  a  mineral  colored  by 
certain  organic  substances  distributed  in  dilute  form;  the  same 
property  is  exhibited  also  by  the  so-called  uranium  glass.  In  the 
two  latter  cases,  as  in  the  first,  it  is  a  matter  of  a  solution  of 
the  fluorescing  substance,  but  in  a  solid  body.  Uranium  glass  and 
the  crystals  of  fluor-spar  are  singly  refracting  media,  hence  emit- 
ting light  whose  color  is  independent  of  the  direction  of  the  vibra- 
tions. Doubly  refracting  fluorescent  crystals  must  obviously 
behave  otherwise;  and  here  two  cases  must  be  distinguished:  — 

i.  When  the  crystal  is  strongly  pleochroic,  the  color  must  of 
course  vary  with  the  direction;  therefore  in  the  case  of  the 
optically  uniaxial  crystals  it  must  vary  with  the  angle  that  the 
vibration  direction  of  the  incident  light  excited  by  the  fluores- 
cence forms  with  the  optic  axis  of  the  crystal.  According  to 
Lommel's  observations  a  substance  exhibiting  this  phenomenon 
particularly  well  is  the  already-mentioned  magnesium  platino- 
cyanide.  When  sunlight  that  has  passed  through  a  blue  or  a- 
violet  glass  is  caused  to  fall  on  the  crystal  face  that  lies  normal 


252  ABSORPTION   OF   LIGHT   IN   CRYSTALS 

to  the  axis,  the  face  in  question  shines  with  scarlet-red  fluorescent 
light,  in  whatever  direction  the  entering  ray  be  polarized;  but 
if  the  light  falls  on  a  face  parallel  to  the  axis,  this  face  exhibits 
"dichroic  fluorescence",  the  light  emitted  being  orange-yellow 
or  scarlet-red,  according  as  the  vibration  direction  of  the  excit- 
ing light  is  parallel  or  perpendicular  to  the  axis.  The  same  two 
colors  are  observed,  of  course,  if  instead  of  polarizing  the  inci- 
dent light  we  cause  the  light  emitted  from  the  crystal  to  pass 
through  a  nicol,  and  hold  this  with  its  vibration  direction  once 
parallel  and  then  perpendicular  to  the  principal  section;  the 
same  colors  are  seen  also  if  we  observe  the  crystal  through  the 
Haidinger  lens.  In  violet  light  the  green  surface  color  (see 
p.  249)  disappears. 

2.  When  the  crystal  is  colorless  and  accordingly  not  dichroic, 
then,  from  a  point  of  its  interior  at  which  fluorescent  light  is 
excited,  there  must  be  propagated  each  of  two  rays  that  on 
emerging  have  equal  intensity.  If  the  two  rays  exhibit  a  dif- 
ference in  intensity,  this  circumstance  proves  that  the  vibrations 
of  the  particles  exciting  the  light  are  favored  in  one  direction. 
Now  a  research  by  Sohncke  shows  that  in  calcite  the  fluoresc- 
ing  particles  vibrate  in  the  greater  degree  parallel  to  the  axis; 
in  apatite,  perpendicular  to  the  axis. 

PHOSPHORESCENCE 

In  the  case  of  certain  substances  in  which,  under  the  in- 
fluence of  irradiation,  an  emission  of  light  takes  place,  the  latter 
continues  even  after  the  irradiation  has  ceased;  in  other  words, 
such  bodies  still  shine  for  a  time  in  the  dark.  This  phenom- 
enon, called  phosphorescence,  may  be  called  forth  also  by  rubbing 
or  by  heating.  Since  phosphorescence  occurs  likewise  with  crys- 
tallized and  doubly  refracting  substances,  we  must  assume  that 
it,  too,  is  subject  to  the  laws  of  pleochroism;  as  to  this,  however, 
no  researches  have  yet  been  made;  and  in  most  cases,  withal, 
the  light  emitted  has  only  a  very  slight  intensity. 


THERMAL   PROPERTIES  253 


INFLUENCE  OF  OTHER  PROPERTIES  ON  THE 
OPTICAL  PROPERTIES  OF  CRYSTALS* 

THERMAL  PROPERTIES 

When  solids  are  heated  a  change  takes  place  in  the  distance 
between  their  smallest  particles;  therefore  with  a  different  tem- 
perature the  forces  which  the  particles  exert  on  one  another,  and 
by  whose  equilibrium  the  distance  between  the  particles  is  deter- 
mined, must  have  become  different.  Since,  then,  the  lumin- 
iferous  ether  of  these  bodies  stands  under  the  influence  of  the 
forces  mentioned,  this  ether  likewise  must  undergo  a  change  by 
the  heat.  As  a  matter  of  fact,  it  is  taught  by  observation  that 
on  a  variation  of  the  temperature  of  a  solid  body  the  trans- 
mission velocity  of  light  in  the  body  becomes  different,  different 
in  such  a  way  that  in  some  solids  the  refractive  index  increases 
with  the  temperature,  while  in  the  majority  of  those  investigated 

*  [For  the  purposes  of  this  partial  translation  it  has  seemed  inexpedient  to  re- 
tain the  classification  of  crystal  properties  described  in  the  Introduction  (pp.  3-8), 
which  is  followed  throughout  Part  I  of  the  original  work  (in  the  fourth  edition) , 
this  classification,  so  far  as  the  present  translation  is  concerned,  would  be  as 
follows: 

A.  Bi-vector  Properties  of  Higher  Symmetry 

(Ellipsoidal  Properties} 

1.  Optical  Properties 

2.  Thermal  Properties  (Influence  of  Heat) 

3.  Homogeneous  Strains 

B.  Bi-vector  Properties  of  Lower  Symmetry 

i.  Elasticity-properties  (Influence  of  Elastic  and  Other  Strains,  and  Op- 
tically Anomalous  Crystals) 

C.  Vector  Properties 

1.  Polar  Piezo-electricity 

2.  Gliding 

3.  Growth  (Twinning) 

But  it  is  believed  that,  since  the  other  crystal  properties  are  here  considered  only 
with  reference  to  their  influence  on  the  optical,  the  classification  in  the  following 
pages,  which  is  similar  to  that  in  the  third  German  edition,  will  be  more  appro- 
priate. The  text,  except  for  the  necessary  rearrangement,  with  the  change  in  form 
thus  entailed,  is  that  of  the  fourth  edition.] 


254  INFLUENCE   OF   OTHER   PROPERTIES 

(as  in  all  the  liquids)  the  contrary  is  the  case.*  Which  of  the  two 
phenomena  occurs,  increase  or  decrease,  depends,  according  to 
the  electro-magnetic  theory  of  light,  on  the  manner  of  the  absorp- 
tion of  light  in  the  body:  most  substances  possess  in  the  ultra- 
violet a  range  of  metallic  reflection,  and  on  heating  the  body 
this  range  is  shifted  toward  the  visible  part  of  the  spectrum;  by 
reason  of  this  shifting  there  is  effected,  at  the  time  when  the 
range  of  metallic  reflection  lies  near  the  middle  of  the  spectrum, 
an  increase  in  the  refraction  and  dispersion,  while  the  refraction 
must,  in  itself,  with  rising  temperature  become  less.f 

Since  the  expansion  of  crystals  by  heat  is  different  with  each 
of  the  five  groups  mentioned  on  page  196, t  these  different  groups 
must,  with  reference  to  the  changes  in  their  optical  relations 
effected  by  the  expansion,  be  treated  separately, — as  will  be 
done  in  the  following. 

i.  SINGLY  REFRACTING  CRYSTALS.  —  These,  as  follows  from 
the  constancy  of  their  angles  for  all  temperatures,  and  as  was 
proved  directly  by  Fizeau's  exact  measurements,!  have  the  same 
coefficient  of  expansion  in  all  directions;  consequently,  by 
heating  such  a  crystal  the  transmission  velocity  of  light  in  it 
is  varied  equally  in  all  directions.  So  soon,  therefore,  as  the 
crystal  has  assumed  the  higher  temperature  uniformly  in  all 
its  parts,  its  refractive  index,  while  smaller!  than  before,  has 
the  same  value  in  all  directions:  THE  CRYSTAL  HAS  REMAINED 

OPTICALLY  ISOTROPIC  AND  REMAINS  SO   AT   ALL  TEMPERATURES. 

When   a   singly  refracting  crystal   exhibits   rotation   of   the 

*  On  account  of  this  variation,  in  exact  determinations  of  the  refractive 
indices  of  a  body  its  temperature  during  the  measurement  must  always  be  stated. 

t  For  particulars,  see  the  inaugural  dissertation  by  J.  Konigsberger:  "Uber 
die  Absorption  des  Lichtes  in  festen  Korpern".  Freiburg,  1900. 

J  Cf.  Phys.  Kryst.  4th  ed.  181-189;  3rd  ed.  169-179. 

§  See  Phys.  Kryst.  4th  ed.  190;  3rd  ed.  181-182. 

||  At  least,  this  is  the  case  with  the  four  substances  investigated  by  Stefan, 
viz.  potassium  chloride,  sodium  chloride  (of  these  the  refractive  index  varies 
greatly  with  the  temperature),  fluor-spar  (calcium  fluorid),  and  alum;  while  the 
diamond  (according  to  A.  Sella)  and  the  amorphous  glass  behave  in  the  opposite 
way. 


THERMAL   PROPERTIES  255 

polarization  plane,  as  is  the  case  with  sodium  chlorate,  for  ex- 
ample (see  p.  222),  its  rotatory  power  varies  with  the  temper- 
ature, but  equally  in  all  directions.  According  to  Sohncke,  with 
rising  temperature  the  rotatory  power  of  the  salt  named  exhibits 
a  considerable  increase. 

2.  UNIAXIAL  CRYSTALS.  —  Exact  determinations  of  the  vari- 
ation of  the  refractive  indices  of  uniaxial  crystals  by  heat  lie  at 
hand  only  on  quartz,  calcite,  beryl,  and  phenacite.  As  for  the 
first  of  these  crystals,  Fizeau  demonstrated  that  when  its  tem- 
perature is  raised  the  refractive  index  both  of  the  ordinary  and 
of  the  extraordinary  ray  becomes  smaller,  the  variation  being 
very  nearly  equal  with  the  two  rays.  For  calcite,  on  the  other 
hand,  he  found  that  both  indices  increase,  the  index  of  the 
ordinary  ray  very  little,  but  the  index  of  the  extraordinary  very 
considerably  An  increase  of  the  refractive  indices,  but  one  that 
was  greater  for  co  than  for  e,  followed  from  the  measurements 
of  Dufet  and  Offret  on  beryl  (emerald).  (This  mineral  takes  an 
exceptional  position  with  respect  also  to  expansion  by  heat.)* 
In  phenacite,  finally,  Offret  found  for  w  and  s  an  increase  that 
was  approximately  the  same  for  both  indices.  According  to 
this,  with  rising  temperature  the  double  refraction  becomes  in 
the  case  of  calcite  considerably  stronger;  in  that  of  quartz  and 
of  beryl,  on  the  other  hand,  weaker. 

From  the  behavior  of  the  uniaxial  crystals  on  heating,  we 
know  that  in  all  directions  forming  equal  angles  with  the  optic 
axis  they  expand  equally;  it  is  therefore  to  be  expected  that  in 
all  such  directions  the  variation  of  the  light  velocity  by  heat 
would  be  the  same,  whether  that  velocity  increase  or  diminish  with 
the  temperature.  Then  A  CRYSTAL  THAT  is  UNIAXIAL  AT  ONE 

TEMPERATURE  MUST  BE  UNIAXIAL  AT  EVERY  OTHER  TEMPERA- 
TURE; and  this  is  confirmed  by  the  observations  made  on  all  the 
numerous  crystals  whose  behavior  toward  light  has  been  investi- 
gated. Since  the  expansion  perpendicular  to  the  optic  axis  has 
a  value  different  from  the  value  parallel  to  the  axis,  so  also  is 

*  Cf.  Phys.  Kryst.  4th  ed.  181;  3rd  ed.  169. 


254  INFLUENCE   OF   OTHER   PROPERTIES 

(as  in  all  the  liquids)  the  contrary  is  the  case.*  Which  of  the  two 
phenomena  occurs,  increase  or  decrease,  depends,  according  to 
the  electro-magnetic  theory  of  light,  on  the  manner  of  the  absorp- 
tion of  light  in  the  body:  most  substances  possess  in  the  ultra- 
violet a  range  of  metallic  reflection,  and  on  heating  the  body 
this  range  is  shifted  toward  the  visible  part  of  the  spectrum;  by 
reason  of  this  shifting  there  is  effected,  at  the  time  when  the 
range  of  metallic  reflection  lies  near  the  middle  of  the  spectrum, 
an  increase  in  the  refraction  and  dispersion,  while  the  refraction 
must,  in  itself,  with  rising  temperature  become  less.f 

Since  the  expansion  of  crystals  by  heat  is  different  with  each 
of  the  five  groups  mentioned  on  page  1964  these  different  groups 
must,  with  reference  to  the  changes  in  their  optical  relations 
effected  by  the  expansion,  be  treated  separately, — as  will  be 
done  in  the  following. 

i.  SINGLY  REFRACTING  CRYSTALS.  —  These,  as  follows  from 
the  constancy  of  their  angles  for  all  temperatures,  and  as  was 
proved  directly  by  Fizeau's  exact  measurements^  have  the  same 
coefficient  of  expansion  in  all  directions;  consequently,  by 
heating  such  a  crystal  the  transmission  velocity  of  light  in  it 
is  varied  equally  in  all  directions.  So  soon,  therefore,  as  the 
crystal  has  assumed  the  higher  temperature  uniformly  in  all 
its  parts,  its  refractive  index,  while  smaller  ||  than  before,  has 
the  same  value  in  all  directions:  THE  CRYSTAL  HAS  REMAINED 

OPTICALLY  ISOTROPIC  AND  REMAINS  SO   AT  ALL  TEMPERATURES. 

When   a   singly   refracting  crystal   exhibits  rotation   of   the 

*  On  account  of  this  variation,  in  exact  determinations  of  the  refractive 
indices  of  a  body  its  temperature  during  the  measurement  must  always  be  stated. 

f  For  particulars,  see  the  inaugural  dissertation  by  J.  Konigsberger:  "Uber 
die  Absorption  des  Lichtes  in  festen  Korpern".  Freiburg,  1900. 

J  Cf.  Phys.  Kryst.  4th  ed.  181-189;  3rd  ed.  169-179. 

§  See  Phys.  Kryst.  4th  ed.  190;  3rd  ed.  181-182. 

II  At  least,  this  is  the  case  with  the  four  substances  investigated  by  Stefan, 
viz.  potassium  chloride,  sodium  chloride  (of  these  the  refractive  index  varies 
greatly  with  the  temperature),  fluor-spar  (calcium  fluorid),  and  alum;  while  the 
diamond  (according  to  A.  Sella)  and  the  amorphous  glass  behave  in  the  opposite 
way. 


THERMAL   PROPERTIES  255 

polarization  plane,  as  is  the  case  with  sodium  chlorate,  for  ex- 
ample (see  p.  222),  its  rotatory  power  varies  with  the  temper- 
ature, but  equally  in  all  directions.  According  to  Sohncke,  with 
rising  temperature  the  rotatory  power  of  the  salt  named  exhibits 
a  considerable  increase. 

2.  UNIAXIAL  CRYSTALS.  —  Exact  determinations  of  the  vari- 
ation of  the  refractive  indices  of  uniaxial  crystals  by  heat  lie  at 
hand  only  on  quartz,  calcite,  beryl,  and  phenacite.  As  for  the 
first  of  these  crystals,  Fizeau  demonstrated  that  when  its  tem- 
perature is  raised  the  refractive  index  both  of  the  ordinary  and 
of  the  extraordinary  ray  becomes  smaller,  the  variation  being 
very  nearly  equal  with  the  two  rays.  For  calcite,  on  the  other 
hand,  he  found  that  both  indices  increase,  the  index  of  the 
'ordinary  ray  very  little,  but  the  index  of  the  extraordinary  very 
considerably  An  increase  of  the  refractive  indices,  but  one  that 
was  greater  for  co  than  for  e,  followed  from  the  measurements 
of  Dufet  and  Offret  on  beryl  (emerald).  (This  mineral  takes  an 
exceptional  position  with  respect  also  to  expansion  by  heat.)* 
In  phenacite,  finally,  Offret  found  for  a>  and  s  an  increase  that 
was  approximately  the  same  for  both  indices.  According  to 
this,  with  rising  temperature  the  double  refraction  becomes  in 
the  case  of  calcite  considerably  stronger;  in  that  of  quartz  and 
of  beryl,  on  the  other  hand,  weaker. 

From  the  behavior  of  the  uniaxial  crystals  on  heating,  we 
know  that  in  all  directions  forming  equal  angles  with  the  optic 
axis  they  expand  equally;  it  is  therefore  to  be  expected  that  in 
all  such  directions  the  variation  of  the  light  velocity  by  heat 
would  be  the  same,  whether  that  velocity  increase  or  diminish  with 
the  temperature.  Then  A  CRYSTAL  THAT  is  UNIAXIAL  AT  ONE 

TEMPERATURE  MUST  BE  UNIAXIAL  AT  EVERY  OTHER  TEMPERA- 
TURE; and  this  is  confirmed  by  the  observations  made  on  all  the 
numerous  crystals  whose  behavior  toward  light  has  been  investi- 
gated. Since  the  expansion  perpendicular  to  the  optic  axis  has 
a  value  different  from  the  value  parallel  to  the  axis,  so  also  is 

*  Cf.  Phys.  Kryst.  4th  ed.  181;  3rd  ed.  169. 


256  INFLUENCE    OF    OTHER    PROPERTIES 

the  variation  that  the  optical  behavior  suffers  in  the  former 
direction/  by  heat,  more  or  less  different  from  the  variation 
it  undergoes  in  the  latter  direction;  in  other  words,  with  the 
crystal  at  a  higher  temperature  the  difference  in  its  optical 
behavior  for  vibrations  parallel  and  perpendicular  to  the  axis, 
and  thus  the  strength  of  its  double  refraction  (see  the  examples  on 
p.  255),  is  greater  or  less.  If  the  latter  is  the  case,  and  if  the 
birefringence  even  at  ordinary  temperatures  is  very  low,  then 
there  is  a  temperature  at  which,  for  one  color,  the  ordinary  and 
the  extraordinary  ray  within  the  crystal  have  equal  velocity;  but 
this  applies  only  to  the  light  of  a  definite  vibration  period,  and 
the  crystal  has  not  ceased  by  reason  of  the  mentioned  equality 
to  be  optically  uniaxial.  (Cf.  footnote  p.  125.) 

Accordingly,  in  consequence  of  a  uniform  increase  of  temper- 
ature the  interference  phenomena  of  uniaxial  crystals  can  expe- 
rience no  change  other  than  such  as  results  from  a  variation 
in  the  refractive  indices  and  in  the  birefringence.  So  the  color 
rings  produced  by  a  plate  cut  normal  to  the  axis  will  become 
either  narrower  or  wider,  but  will  always  retain  their  circular  form. 

Like  the  ordinary,  so  also  does  the  circular  double  refraction 
experience  a  variation  with  the  temperature.  This  fact  stands 
in  accord  with  the  assumption,  mentioned  on  page  218,  that  the 
rotatory  power  of  the  crystals  in  question  depends  on  a  spiral 
grouping  of  doubly  refracting  lamellae;  for,  since  the  double 
refraction  of  these  lamellae  is  influenced  by  the  temperature,  the 
same  must  be  the  case  with  the  rotation  of  the  polarization  plane 
resulting  from  this  double  refraction.  It  has  been  demon- 
strated in  the  case  of  quartz  that  its  rotatory  power  increases 
with  rising  temperature,  increasing  faster  than  the  temperature 
but,  perceptibly,  for  all  colors  the  same  amount. 

3.  BIAXIAL  CRYSTALS.  —  In  these  crystals  the  expansion  by 
heat  is  not  equal  in  the  vibration  direction  of  the  greatest,  in 
that  of  the  intermediate,  and  in  that  of  the  least  light  velocity; 
consequently,  when  the  crystal  is  brought  to  a  higher  temper- 
ature the  three  principal  refractive  indices  experience  unequal 


THERMAL   PROPERTIES  257 

variation.  With  aragonite,  where  by  Rudberg's  measurements 
this  fact  was  first  demonstrated,  there  corresponds  to  the  direc- 
tions of  the  greatest,  the  intermediate,  and  the  least  expansion 
by  heat  the  least,  the  greatest,  and  the  intermediate  decrease 
of  the  refractive  indices;  in  the  case  of  gypsum,  according  to 
Dufet's  observations,  to  the  directions  of  the  greatest,  the  inter- 
mediate, and  the  least  expansion  there  correspond  the  greatest, 
the  intermediate,  and  the  least  decrease  in  the  refraction  of  the 
rays  vibrating  parallel  to  those  directions,  except  that  here  the 
directions  of  the  maxima  and  minima  for  different  colors  do  not 
exactly  coincide,  because  gypsum  belongs  with  the  crystals  that 
have  their  principal  vibration  directions  dispersed.  The  three 
isomorphous  minerals  barite  (heavy  spar,  barium  sulphate), 
celestite  (strontium  sulphate),  and  anglesite  (lead  sulphate), 
according  to  the  researches  of  Arzruni,  all  exhibit  the  most 
marked  decrease  for  the  largest  refractive  index  7-,  the  least 
for  a;  but  there  is  no  such  agreement  in  the  relative  expansion 
by  heat  along  the  three  principal  vibration  directions;  for, 
while  with  the  first  of  these  bodies  the  direction  of  greatest  ex- 
pansion is  that  of  the  intermediate  light  velocity,  with  angle- 
site  it  is  in  the  direction  of  the  least  light  velocity  that  the  crystal 
expands  the  most.  For  barite  somewhat  different  values  were 
found  by  Offret,  who  showed  besides,  in  a  very  comprehensive 
and  careful  research,  that  in  topaz,  cordierite,  sanidine,  and 
oligoclase  the  three  principal  refractive  indices  do  not  diminish 
but  increase.  A  decrease  of  the  refractive  indices  was  exhibited 
only  by  those  substances  that  have  very  large  coefficients  of  ex- 
pansion. It  may  therefore  be  assumed  that  here  the  decrease  is 
caused  by  the  great  change  of  volume,  i.e.  diminution  of  density, 
resulting  from  the  rise  in  temperature,  and  that  the  refringency 
proper  to  the  substance,  referred  to  the  same  density,  always 
increases  with  the  temperature.  Offret's  research  embraced  also 
the  variation  of  the  birefringence  and  of  the  dispersion  by  heat, 
and  led  to  the  following  result:  With  all  the  bodies  investigated 
the  dispersion  increases  with  the  temperature;  the  double  re- 


258  INFLUENCE   OF   OTHER   PROPERTIES 

fraction,  on  the  other  hand,  not  only  in  different  substances  but 
also  for  the  different  principal  refractive  indices  of  one  and  the 
same  substance,  increases  in  part,  in  part  grows  less. 

That  with  every  biaxial  crystal  the  variation,  produced  by 
heat,  in  the  light  velocities  corresponding  to  the  three  principal 
vibration  directions  is  different  for  each  of  these  directions,  may 
easily  be  proved  in  an  indirect  way;  and  for  numerous  sub- 
stances this  proof  has  already  been  supplied.  For  if  the  three 
principal  refractive  indices  are  varied  unequally  by  heat,  so  too 
is  there  a  change  in  tneir  ratios  to  one  another;  but  on  these 
ratios  depends  the  size  of  the  optic  axial  angle,  and  accordingly 
this  angle  likewise  must  be  a  function  of  the  temperature,  be- 
coming larger  or  smaller  when  the  crystal  is  heated.  To  verify 
this,  one  must  employ  the  method  described  on  page  187,  with 
a  modification  such  that  during  the  measurement  of  its  axial 
angle  the  crystal  is  at  a  constant  higher  temperature.  This 
latter  is  accomplished  by  inserting  between  condensing  lens 
and  objective  of  the  horizontal  polariscope  a  metal  box  project- 
ing some  distance  on  both  sides;  and  in  both  the  front  and 
the  back  wall  of  this  box  there  is  fitted  a  plane-parallel  glass 
plate,  so  that,  as  before,  light  may  be  caused  to  fall  through 
the  instrument.  Hence,  if  between  these  two  glass  plates,  in- 
side the  box,  the  crystal  is  so  fastened  that  it  is  centered  and 
rotatable,  and  if  the  air  in  the  box  is  heated  and  maintained 
throughout  a  long  time  at  constant  temperature  (with  the  aid 
of  an  inserted  thermometer),  — whereby  the  crystal  also  has 
assumed  that  same  temperature  in  all  its  parts, —  then  the 
measurement,  arranged  just  as  at  ordinary  temperatures,  gives 
us  the  size  of  the  axial  angle  corresponding  to  that  higher 
temperature. 

Now  numerous  determinations  of  the  optic  axial  angle  at 
different  temperatures,  made — chiefly  by  Des  Cloiseaux — with 
the  aid  of  a  heating  apparatus  of  this  kind,*  have  shown  that 
as  a  matter  of  fact  the  angle  in  question  varies  with  the  tem- 

*  Cf.  footnote  f,  P-  30. 


THERMAL   PROPERTIES  259 

perature  in  all  the  bodies  investigated.  In  the  case  of  some 
the  variation  is  so  slight  that  the  difference  could  hardly  be  veri- 
fied by  the  measurement;  with  the  greater  number  the  angle 
changed  several  degrees  when  the  temperature  was  raised  100°; 
while  there  are  also  crystals,  finally,  whose  optic  axes  vary  their 
direction  many  degrees,  even  when  heated  but  slightly.  Among 
these  latter  crystals  is  included  gypsum,  for  example,  whose 
axial  angle  diminishes  so  fast  when  the  crystal  is  heated  that 
even  at  a  temperature  below  100°  C.  it  is  zero;  thus,  at  a  cer- 
tain temperature  the  smallest  refractive  index  becomes  equal  to 
the  intermediate.  The  crystal  is  then  temporarily  uniaxial,  but 
of  course,  on  account  of  the  dispersion  of  the  axes,  at  one  tem- 
perature it  is  uniaxial  only  for  one  color,  not  for  the  others  (while 
a  really,  i.e.  permanently,  uniaxial  crystal  is,  as  we  know,  uniaxial 
for  all  colors  and  for  all  temperatures).  If  a  gypsum  crystal  is 
still  further  heated,  so  that  the  previously  smallest  refractive 
index  increases  still  more,  thus  becoming  greater  than  the  pre- 
viously intermediate,  the  axial  plane  is  now  perpendicular  to  its 
former  position  (in  other  words,  on  further  heating,  the  optic 
axes  spread  out  in  the  plane  normal  to  the  former  axial  plane) ; 
and  one  easily  sees  that,  for  the  same  color  for  which  their 
angle,  as  compared  with  their  angle  for  the  other  colors,  had 
previously  the  smallest  value,  the  axes  now  include  a  greater 
angle  than  for  any  other  color.  The  same  phenomenon  is  ex- 
hibited by  sanidine  and  by  glauberite;  with  the  latter  mineral, 
whose  crystals  have  a  very  small  axial  angle,  the  uniaxial 
state,  arising  for  the  different  colors  in  succession,  exists  for 
violet,  blue,  and  so  on  to  red,  light  within  the  small  range  of 
temperature  from  17°  to  58°  C. 

The  last-named  substances  belong,  besides,  to  one  of  the 
groups  of  crystals  whose  principal  vibration  directions  for  differ- 
ent colors  do  not  coincide,  wherefore  the  principal  thermic  axes,* 
also,  have  an  orientation  different  from  that  of  the  principal 

*  [These  are  defined  by  the  author  elsewhere  as  the  only  crystallographic 
directions  that  remain  mutually  perpendicular  at  all  temperatures.] 


26o  INFLUENCE   OF   OTHER   PROPERTIES 

vibration  directions.  In  agreement  with  this,  in  these  crystals 
the  direction  of  the  acute  bisectrix,  likewise,  is  dependent  not 
only  on  the  color  but  also  on  the  temperature;  that  is,  with 
rising  temperature  there  is  a  change  in  the  orientation  of  the 
bisectrix  as  referred  to  the  crystal  plate,  and  consequently  in  the 
field  of  view  one  observes  an  unequal  angular  movement  of 
the  two  optic  axes.  For  example,  if  we  gradually  heat  a  gypsum 
plate  cut  perpendicular  to  the  acute  bisectrix  for  intermediate 
colors,  we  see  distinctly  that  the  two  axes  approach  each  other 
with  unequal  velocity,  until,  when  the  temperature  reaches  a 
certain  point,  they  coincide  in  a  direction  that  stands  percep- 
tibly oblique  to  the  plate;  on  further  heating,  the  two  axes  re- 
cede from  each  other  in  the  plane  that  stands  perpendicular  to 
the  former  axial  plane,  receding  equally  fast,  —  because  in  gyp- 
sum one  principal  optic  section  is  common  to  all  colors,  where- 
fore with  respect  to  this  plane  all  optical  phenomena  take  place 
with  absolute  symmetry, — but  the  position  of  the  bisectrix  is 
shifted  farther  in  the  same  sense  as  before.  In  crystals  where 
all  three  principal  vibration  directions  for  all  colors  have  iden- 
tical orientation,  there  exists,  for  all  temperatures,  complete 
symmetry  of  the  optical  properties  with  reference  to  the  three 
mutually  perpendicular  principal  sections  (so  far  as  the  sub- 
stance remains  unaltered).  Such  a  crystal,  therefore,  subjected 
to  the  same  experiment,  exhibits  a  mutual  approach  or  recession 
of  the  optic  axes  that  is  absolutely  equal  in  rate  for  the  two 
axes;  in  other  words,  it  exhibits  constant  orientation  of  the 
bisectrix. 

Hence,  from  the  phenomena  just  described  it  follows  that,  in 
consequence  of  the  variation  of  the  double  refraction  with  the  tem- 
perature, when  the  heat  content  of  a  biaxial  crystal  is  different  the 
index-surface  of  the  crystal  has  a  different  form  (i.e.  different 
axial  ratios) ;  but  that  its  crystallographic  orientation  is  constant, 
provided  there  exists  no  dispersion  of  the  axes  for  the  differ- 
ent colors.  Otherwise  its  orientation,  also,  depends  on  the 
temperature. 


HOMOGENEOUS   STRAIN  261 

ELASTIC  STRAIN  BY  MECHANICAL  FORCES 
Homogeneous    Strain 

The  expansion  of  crystals  by  heat  is  to  be  regarded  as  a 
special  case  of  "  homogeneous  strain",  or  "  homogeneous  defor- 
mation ".  By  such  a  strain,  or  deformation,  we  understand  a 
change  in  the  shape  of  a  homogeneous  body  by  which  the  body 
does  not  cease  to  fulfill  the  conditions  of  homogeneity  (see  p.  3), 
and  for  which  the  following  relations  exist  between  the  origi- 
nal body  and  the  strained  body:  First,  any  straight  line  in  the 
unstrained  body  is  straight  likewise  after  the  strain,  while  by 
the  strain  the  direction  of  that  line  in  general  surfers  a  vari- 
ation; second,  every  two  straight  lines  that  are  parallel  in  the 
original  body  are  parallel  also  in  the  strained  body,  whatever 
change  may  have  taken  place  in  their  direction.* 

The  change  produced  by  a  homogeneous  deformation,  in  the  linear 
dimensions  of  a  crystal,  necessitates  a  displacement  of  the  crystal  particles 
from  the  position  where  the  forces  acting  among  them  are  in  equilibrium. 
This  calls  for  a  certain  amount  of  work.  In  the  case  of  thermic  expansion 
this  work  is  done  by  the  heat  added ;  in  the  other  case  it  requires  some  exter- 
nal mechanical  force.  When  the  latter  ceases  to  act  the  particles  resume,  in 
consequence  of  the  above-mentioned  internal  forces  of  the  crystal,  the  rela- 
tive position  that  corresponds  to  equilibrium  of  those  forces;  that  is,  when  the 
strain  produced  by  the  external  force  does  not  exceed  a  certain  limit.  This 
power  of  a  solid  body  to  resume  its  former  shape  is  called  "elasticity";  and 
accordingly  those  changes  in  a  solid  that,  in  consequence  of  its  elasticity,  dis- 
appear when  the  straining  forces  cease  to  act,  are  designated  as  "elastic 
strains".  The  value  which  the  external  force  must  not  exceed  if  no  perma- 
nent alteration  in  the  shape  of  the  body  is  to  arise,  is  termed  the  "elastic 
limit".  Within  this  limit  the  displacement  of  the  particles,  and  therefore 
the  deformation  of  the  body,  is  proportional  to  the  forces  acting  on  the  body. 

*  From  these  relations  it  follows  directly  that  a  plane  appearing  as  a  boundary- 
face  of  the  original  body  is,  after  the  strain  of  the  body,  still  a  plane,  but  one 
whose  position,  i.e.  whose  angle  with  other  planes  bounding  the  body,  can  have^ 
experienced  a  variation;  and  it  follows,  further,  that  every  two  parallel  planes 
bounding  the  body  are,  after  the  strain,  likewise  parallel,  in  whatever  manner  or 
degree  their  position  may  have  changed. 


262  INFLUENCE   OF   OTHER   PROPERTIES 

When  the  external  force  acting  on  a  crystal  is  so  great  that  the  elastic 
limit  is  overstepped,  deformations  generally  arise  which  are  not  homogeneous; 
and  if  the  "  limit  of  solidity  "  is  exceeded,  i.e.  if  the  external  force  is  so  great 
as  to  completely  separate  the  component  particles,  the  crystal  altogether 
ceases,  as  a  unit,  to  exist. 

Since  the  homogeneous  strains  produced  by  equal  pressure 
or  tension  in  all  directions  correspond  absolutely  to  those  effected 
by  a  uniform  change  in  temperature,  the  exposition  given  on  page 
253  el  seq.  of  the  influence  of  the  latter  strains  on  the  optical 
properties  of  crystals  applies  also  to  the  influence  of  the  former. 
The  changes  they  produce  in  the  velocity  of  light  are  accord- 
ingly equal  in  all  optically  equivalent  directions;  so  that  A  SINGLY 

REFRACTING    CRYSTAL  IS   STILL   SINGLY    REFRACTING    EVEN  AFTER 
THE  STRAIN,  A  UNIAXIAL  CRYSTAL   STILL  UNIAXIAL,  AND  SO  ON. 

Those  mechanical  forces  that  effect  homogeneous  strains  (equal  pressure 
or  tension  from  all  sides)  are  the  only  ones  with  which,  as  with  a  uniform 
change  in  temperature,  the  cause  of  the  strain  is  scalar,  —  i.e.  independent  of 
the  direction.  In  such  cases,  that  the  effect  produced  in  the  crystal  is  differ- 
ent in  different  directions  is  only  because  the  crystal  is  not  constituted  alike 
In  different  directions  and  consequently  does  not  offer  the  same  resistance 
to  deformation.  The  relations  are  quite  different,  and  considerably  more 
complex,  when  the  cause  of  the  strain  is  a  vector  quantity;  in  other  words, 
when  a  mechanical  force  acts  in  a  crystallized  body  in  a  certain  direction. 
The  effect  of  definitely  directed  forces  of  equal  magnitude,  but  of  different 
orientation  as  referred  to  definite  crystal logra phi c  directions,  can  no  longer  be 
represented  in  the  simplest  case  (i.e.  for  a  singly  refracting  crystal)  by  a  sphere, 
nor  in  the  most  general  case  by  a  triaxial  ellipsoid  whose  three  axes  deter- 
mine the  numerical  value  of  the  property  in  question  for  all  the  remaining  direc- 
tions in  the  crystal.  The  crystal  properties  of  elasticity  and  cohesion,  which 
belong  under  this  head,  are  never  equal  in  all  directions.  Moreover,  in  the  case 
of  such  a  deformation  the  crystal  itself  generally  ceases  to  be  homogeneous. 

Elastic  Strain  not  Homogeneous 

The  differences  produced  in  the  constitution  of  a  body  in 
different  directions  by  tension  or  pressure  acting  in  a  definite 
direction,  are  transferred  to  the  constitution  of  the  luminiferous 
ether  standing  under  the  influence  of  the  particles  of  the  body; 


ELASTIC    STRAIN   NOT    HOMOGENEOUS  263 

so  that  vibrations  taking  place  in  directions  that  previously 
were  optically  equivalent  are  now  no  longer  transmitted  by 
the  ether  with  equal  velocity.  The  laws  after  which  these 
variations  take  place  were  investigated  experimentally  by  F. 
Neumann,  and  explained  theoretically  on  the  basis  of  this 
assumption:  that,  when  an  amorphous  body  is  extended  or  com- 
pressed uniformly  at  all  points  in  a  definite  direction,  it  takes  on 
the  properties  of  a  homogeneous  optically  biaxial  crystal  in 
which  the  axes  of  the  index-surface  coincide  with  those  of  the 
ellipsoid  of  strain.*  In  a  body  that  is  extended  or  compressed 
ununiformly,  although  the  above  must  indeed  be  assumed  for 
every  individual  point,  yet  the  form  and  the  orientation  of  the 
optical  index-surface  vary  with  the  locality  in  the  strained 
body.  Since  the  behavior  of  crystals  toward  straining  forces 
is  different  in  different  directions,  the  optical  relations  of  crys- 
tals in  the  strained  state,  even  when  the  crystals  in  question  are 
singly  refracting,  are  very  much  more  complex  than  with  amor- 
phous bodies;  let  us  therefore  treat  first  the  phenomena  to  be 
observed  in  the  latter  bodies. 

Should  we  take  a  cylinder  made  from  ordinary  glass,  and 
compress  it  perpendicularly  to  the  two  circular  basal  faces  in 
such  a  way  that  its  interior  were  straine.d  uniformly  at  all  points, 
it  would  take  on  exactly  the  same  optical  properties  as  are  pos- 
sessed by  a  negative  uniaxial  crystal  whose  optic  axis  lies  in  the 
direction  of  the  pressure.!  By  compression,  therefore,  the  glass 

*  In  a  crystal  subjected  to  a  homogeneous  strain  there  always  exist  three  mu- 
tually perpendicular  directions  whose  position  is  not  altered  by  the  strain;  and 
parallel  to  these  directions  there  take  place  the  greatest,  the  intermediate,  and  the 
least  variation  of  the  linear  dimensions  of  the  crystal.  These  directions  are  desig- 
nated as  the  "principal  axes  of  strain".  The  increase  or  decrease  of  the  linear 
dimensions  along  these  three  principal  axes  is  in  general  different  for  each  axis,  so 
that  a  sphere  made  from  the  crystal  is  transformed  by  the  strain  into  a  triaxial 
ellipsoid  —  the  "  ellipsoid  of  strain  ". 

f  This  effect  of  pressure  can  be  very  beautifully  demonstrated  in  soft  gelatin, 
if  a  circular  plate  of  the  same  is  brought,  between  glass  plates,  upon  the  stage 
of  the  Norrenberg  polariscope  and  the  ocular-tube  screwed  down  until  the  gel- 
atin plate  is  compressed:  the  plate  then  exhibits  the  normal  interference-figure 


264  INFLUENCE    OF   OTHER    PROPERTIES 

assumes  negative  double  refraction.  Precisely  the  same  extension, 
in  the  same  direction,  would  produce  the  opposite  effect,  and  the 
glass  would  behave  like  a  uniaxial  crystal  with  positive  double 
refraction;  by  such  a  strain,  therefore,  the  glass  becomes  posi- 
tively doubly  refracting. 

The  fact  that  in  glass  compression  in  the  direction  of  vibra- 
tion of  a  ray  does  indeed  call  forth  an  increase  of  its  transmis- 
sion velocity,  and  extension  a  decrease,  can  be  proved  with  the 
aid  of  the  arising  phenomena  of  double  refraction,  whose  char- 
acter may  be  determined  by  one  of  the  methods  described  on 
page  198  et  seq.  These  phenomena,  and  both  of  them  (that  pro- 
duced by  pressure  and  that  produced  by  tension)  simultaneously, 
moreover,  can  be  brought  into  view  most  simply  by  means  of 
the  following  experiment,  suggested  by  F.  Neumann:  Two  strips 
of  plate-glass  are  taken,  2  or  3  decimeters  in  length,  about 
\  decimeter  wide,  and  from  5  to  6  mm.  thick,  and  between  them 

is  laid  a  metal  wire;  the  glass 
strips  are  then  gradually  bent  to- 
gether, best  by  means  of  clamp- 
screws,  until  their  ends  touch, 
so  that  the  concave  sides  of  the 
strips  are  turned  toward  each 
other.  (See  Fig.  116,  repre- 
senting the  glass  strips  as  seen 
from  the  side  and  from  above.) 

In  consequence  of  this  bending  the  outer,  or  convex,  faces  of 
the  plates  have  become  longer  than  before,  but  the  inner,  or 

of  a  uniaxial  crystal.  A  mixture  of  wax  and  resin,  pressed  moderately  between 
two  glass  plates,  behaves  in  just  the  same  way;  and  such  a  preparation  exhibits 
the  phenomenon  even  permanently.  However,  many  authors  assume  that  this 
phenomenon  depends  on  the  presence  in  such  organic  substances  of  very  minute 
crystalline  particles:  that  these  particles  lie  irregularly  distributed,  so  that  the 
several  doubly  refracting  effects  which  they  exert  on  the  light  are  mutually  neutral- 
ized, while  by  pressure  they  become  oriented  parallel  to  one  another  and  now  exercise 
a  common  optical  effect.  Moreover,  flint  glass  having  a  high  lead  content  behaves 
differently  from  ordinary  glass,  in  that  by  compression  it  becomes  negatively  doubly 
refracting.  *****) 


ELASTIC    STRAIN   NOT   HOMOGENEOUS  265 

concave,  faces  shorter;  the  former  faces  accordingly  are  stretched 
in  their  long  direction,  while  the  latter  are  compressed  in  that 
same  direction.  This  is  shown  more  clearly  by  Fig.  117,  the 
amount  of  bending  as  here  represented  being  far  greater  than  is 
really  possible  with  strips  of  glass.  In  the  outermost  layer  of 


Fig.  117. 

the  glass,  e.g.  at  a,  the  direction  of  the  extension  is  parallel  to  //; 
at  this  point  therefore  the  least  light  velocity  is  that  of  the  rays 
vibrating  parallel  to  //,  and  the  greatest  (namely,  equal  to  the 
transmission  velocity  in  the  uncompressed  glass)  that  of  the  rays 
vibrating  parallel  to  qq.  At  a  point  6,  lying  farther  inward  in  the 
strip,  there  is  still  a  lengthening,  although  a  less  considerable  one, 
in  the  direction  //;  so  here,  too,  the  vibrations  parallel  to  //  have  the 
least  transmission  velocity  and  those  parallel  to  qq  the  greatest, 
but  the  difference  between  the  two  velocities  is  smaller.  Along 
the  middle  of  the  glass  strip,  i.e.  at  c  and  all  points  of  the  line 
CC',  which  passes  through  c,  the  difference  in  question  will  be 
zero;  this  is  the  line  that  has  the  same  length  as  with  the  glass  strip 
in  the  unbent  state,  the  line  that  has  experienced  neither  a  com- 
pressing nor  a  stretching.  On  the  concave  side,  on  the  other  hand, 
the  glass  strip,  by  reason  of  the  bending,  has  been  compressed  in 
its  long  direction;  and  this  compression  is  less  considerable  along 
the  line  DDf  than  along  77',  the  latter  line  having  become  shorter 
than  any  other  in  the  strip.  Both  at  d  and  at/,  in  consequence 
of  the  compression,  the  transmission  velocity  of  the  rays  vibrating 
parallel  to  I'V  is  the  greatest,  and  that  of  the  rays  vibrating  perpen- 
dicularly to  them,  or  parallel  to  q'q',  the  least;  but  at/ the  differ- 
ence between  the  two  velocities  is  at  its  maximum.  Accordingly 
the  glass  strip  must  be  doubly  refractive  throughout  its  whole 


266  INFLUENCE   OF   OTHER    PROPERTIES 

thickness  as  far  as  a  zone  in  the  middle;  the  double  refrac- 
tion is  strongest  in  the  outermost  and  the  innermost  zone,  but 
in  the  one  zone  negative  and  in  the  other  positive,  while  from 
these  two  extremes  it  diminishes  in  strength  inwards  to  the  neu- 
tral zone  mentioned,  where  it  is  zero.  Hence,  if  we  bring  the 
two  bent  strips  of  glass  between  two  crossed  nicols  (in  the  polar- 
iscope  with  parallel  light)  in  such  a  way  that  the  light  rays  pass 
through  the  strips  parallel  to  the  metal  wire  (shown  in  Fig.  116) 
and  accordingly  through  an  approximately  J  decimeter  thickness 
of  glass,  all  rays  except  those  falling  in  the  neutral  zone  are  split 
up  into  two  polarized  rays,  which  vibrate  parallel  to  //  and  qq 
and  which  are  transmitted  with  different  velocity.  With  the  dimen- 
sions given,  a  degree  of  bending  even  less  than  represented  in  Fig. 
116  suffices,  in  the  inner  and  the  outer  boundary-layer,  to  impart 
to  the  rays  arising  by  the  double  refraction  a  difference  of  path  that 
amounts  to  several  wave  lengths;  in  the  zones  lying  nearer  the 
center  the  path  difference  acquired  is  naturally  smaller.  At  a 
certain  distance  outside  the  neutral  zone  the  path  difference 
will  be  exactly  A;  at  this  distance  therefore,  with  crossed  nicols, 
there  will  occur  total  annihilation  of  the  light,  just  as  in  the 
singly  refracting  zone  itself;  and  the  same  will  take  place  at  the 
distances  where  the  arising  difference  of  path  is  2  A  etc.  In  case 
the  glass  strips  are  placed  with  their  long  direction  parallel  to 
one  of  the  nicols  they  will  naturally  appear  dark  all  over. 
But  with  the  strips  in  any  other  position,  when  the  light  used  is 
homogeneous,  there  must  appear,  between  the  zones  of  o,  X,  zX, 
etc.,  difference  of  path,  zones  of  brightness;  and  the  brightness 
of  these  zones  is  greatest  half-way  between  two  dark  ones,  since 
on  these  intermediate  lines  the  difference  of  path  amounts  to 
£^,  §^,  etc.  The  greatest  difference  in  intensity  between  the 
bright  and  the  dark  zones  is  found,  of  course,  when  the  long 
direction  of  the  glass  strips  forms  45°  with  the  vibration  direc- 
tions of  the  nicols.  Hence  follows  as  a  matter  of  course  the 
phenomenon  arising  in  white  light:  The  neutral  zone  at  the 
center  of  the  glass  strip  is  black  throughout  its  whole  length; 


ELASTIC   STRAIN   NOT  HOMOGENEOUS  267 

on  each  side  of  the  neutral  zone,  parallel  to  it  and  to  the  outer 
faces  of  the  glass  strip,  there  appear  stripes  of  color,  and  the 
colors  of  these  stripes  lie  in  exactly  the  same  sequence  as  in  the 
color  rings  (appearing  in  convergent  light)  of  uniaxial  crystals 
cut  perpendicular  to  the  axis. 

If  we  take  a  circular  plate  of  glass  and  wind  a  strong  cord 
around  it  upon  its  cylindrical  lateral  surface,  then,  if  we  draw 
tightly  on  the  cord,  the  plate  is  pressed  together  from  all  points 
of  the  periphery  toward  the  center;  it  is  thus  compressed  the 
most  at  the  edges,  and  gradually  less  and  less  from  the  edges  to 
the  center.  In  parallel  light  all  points  of  equal  pressure,  i.e.  all 
points  lying  on  the  circumference  of  a  circle  concentric  with  the 
plate,  exhibit  the  same  color;  there  accordingly  appear  the  cir- 
cular isochromatic  curves  with  the  black  cross,  exactly  as  when 
uniaxial  crystals  are  observed  in  convergent  light.* 

If  we  press  together  a  square  piece  of  glass,  not  uniformly 
from  two  opposite  faces  but  only  from  two  points,  and  bring  it 
into  parallel  polarized  light  between  crossed  nicols  in  such  a  way 
that  the  straight  line  connecting  these  two  points  includes  45° 
with  the  nicols,  the  parts  near  the  two  points  of  pressure  become 
lighted  up,  and  with  greater  pressure  a  color  arises.  But,  from 
the  two  points  of  pressure  this  lightening  becomes  less  in  all 
directions;  that  is  to  say,  directly  at  these  points  the  glass  is 
under  the  greatest  compression,  and  the  compressing  stress  dimin- 
ishes with  increasing  distance  from  them;  the  central  portion  is 
not  under  stress  at  all  and  hence  remains  dark.  When  the 
pressure  exerted  is  greater,  however,  the  adjacent  particles  of 
the  glass  will  have  assumed  a  so  much  greater  degree  of  compres- 
sion in  the  direction  of  pressure  than  parallel  to  it  that  the  two 
wave  systems  arising  by  double  refraction  are  displaced,  as  com- 
pared with  each  other,  to  the  amount  of  several  wave  lengths; 

*  A  small  amount  of  double  refraction  is  exhibited  by  lenses  in  optical  instru- 
ments, if  pressure  is  exerted  upon  them  by  their  mounting.  In  a  conoscope  con- 
taining such  lenses  we  can  of  course,  between  crossed  nicols,  obtain  no  uniformly 
dark  field  of  view. 


268 


INFLUENCE   OF  OTHER  PROPERTIES 


--N 


and  accordingly,  at  the  point  of  pressure  there  appears  a  color  of 
the  third  or  the  fourth  order.     Figure  118  shows  the  phenomenon 
then  observed  in  parallel  light  when  the  direction  of  pressure,  ab 
,  (indicated  by  arrows),  is 

parallel  to  the  principal 
section  of  a  nicol:  since 
from  a  and  b  the  com- 
pression diminishes  in  all 
directions,  so  also  does 
the  strength  of  the  double 
refraction,  and  conse- 
quently at  greater  dis- 
tances there  appear  other 
interference  colors;  these 

M'  colors  yield,  together,  a 

figure  made  up   of    iso- 

chromatic    curves,  which    is   very   similar   to    the    well-known 
lemniscate  system  of  biaxial  crystals. 

If  a  piece  of  glass  is  heated,  but  not  uniformly,  then  owing 
to  the  unevenness  of  the  expansion  there  arise  in  it  extensions 
and  compressions,  and  the  corresponding  phenomena  of  double 
refraction  appear.  These  latter  can  be  made  permanent,  more- 
over, by  bringing  the  glass  to  a  high  temperature  and  then 
rapidly  cooling  it;* we  then  obtain  "  cooled  glasses  ",  so  called, 
which  yield  in  polarized  light,  according  to  their  shape,  the  most 
multifarious  interference-figures. 

Among  the  amorphous  organic  bodies  there  are  many  so- 
called  colloid  substances  (as  collodion,  gelatin,  etc.),  and  these 
have  the  property,  in  passing  from  the  dissolved  to  the  solid 
state,  on  removal  of  the  solvent  by  drying,  of  exhibiting  con- 
siderable contraction.  If  such  bodies  are  caused  to  dry  under 
circumstances  such  that  the  bodies  cannot  shrink  uniformly  in 
all  directions,  then  in  the  directions  in  which  the  contraction  was 
hindered  they  must  be  in  a  state  of  strain.  If,  for  example,  a 
strong  solution  of  gelatin  is  poured  while  warm  into  a  frame 


ELASTIC    STRAIN   NOT   HOMOGENEOUS  269 

lying  on  a  glass  plate  (the  frame  should  be  provided  with  a 
handle),  and  if  after  the  congelation  the  resulting  colloid  cake  is 
lifted  from  the  glass  plate  and  allowed  to  dry  freely  in  the  frame, 
then  the  cake  adheres  to  the  frame  all  around  and  accordingly 
cannot  shrink  in  its  own  plane.  After  the  gelatin  has  hardened 
completely  a  plate  thus  formed  is  therefore  in  a  state  of  strain  in 
all  directions  parallel  to  its  plane.  Should  we  succeed  in  mak- 
ing this  strain  equal  in  all  these  directions,  such  a  plate  would 
necessarily  exhibit  in  convergent  light  the  interference- figure  of 
a  uniaxial  crystal;  since,  however,  the  gelatin  does  not  dry  uni- 
formly in  all  its  parts,  and  many  times  even  comes  loose  from 
the  rim  in  places,  it  always  happens,  in  fairly  large  plates,  that 
only  single  parts  have  remained  in  a  sufficiently  uniform  state 
of  strain  to  exhibit  the  phenomenon  mentioned.  Commercial 
gelatine  plates,  such  as  are  used  by  lithographers,  sometimes 
exhibit  the  dark  cross  distinctly,  and  when  several  of  them  are 
arranged  one  above  the  other  the  color  rings  also  appear;  in 
these  plates  the  double  refraction  is  negative,  whence  it  follows 
that  tension  has  just  the  same  influence  in  gelatin  as  in  glass. 
But  with  plates  congealed  in  a  frame,  on  the  other  hand,  the 
strain  in  different  directions  parallel  to  the  plate  is  in  general 
unequal;  so  the  plate  must  assume  the  properties  of  a  biaxial 
crystal.  As  a  matter  of  fact,  every  part  of  a  plate  of  this  sort  in 
which  the  strain  remains  nearly  constant  over  a  certain  extent 
of  the  plate,  exhibits  in  convergent  light  the  interference-figure 
of  the  biaxial  crystals;  but  the  figures  yielded  by  different  parts  of 
one  and  the  same  plate  have  not  the  same  axial  angle,  nor  their 
axial  planes  the  same  direction,  because  at  the  various  points 
of  such  a  plate  there  is  a  diversity  in  the  relative  amount  of 
strain  in  different  directions  and  in  the  orientation  of  the  maxi- 
mum strain.  Finally,  the  distribution  of  the  strain  naturally 
depends  on  the  form  of  the  frame  in  which  the  congelation 
took  place.  In  the  case  of  thin  plates,  because  of  the  double 
refraction  being  small  in  amount,  the  interference  phenomenon 
becomes  reduced  to  the  appearance,  on  a  light  background,  of 


270  INFLUENCE   OF   OTHER  PROPERTIES 

the  dark  cross  or  the  dark  hyperbolas  (according  as  the  optic 
axial  plane  at  the  point  in  question  lies  parallel  to  one  of  the 
nicols  or  oblique  to  both);  therefore  one  observes  only  the 
central  part,  as  it  were,  of  the  phenomenon  represented  in 
Fig's  836  and  846.  But  with  thicker  plates  the  color  rings, 
likewise,  appear  in  the  field  of  view. 

Similar  behavior  is  exhibited  by  horn,  animal  bladder  (es- 
pecially when  in  several  layers),  and  other  organic  substances. 
Finally,  also,  in  the  above-mentioned  cooled  glasses  there  are 
places  that  give  distinct  axial  figures;  and  where  the  axial  angle 
is  rather  large  these  figures  manifest  a  distinct  dispersion  of 
the  axes,  whence  it  must  be  concluded  that  by  pressure  the 
transmission  velocity  of  the  light  of  different  colors  is  influenced 
unequally. 

Like  the  transmission  velocity,  so  too  is  the  absorption  of 
light  influenced  by  strain.  In  consequence  of  this  a  pressure 
or  tension  that  transforms  an  amorphous  body  into  a  doubly 
refracting  one  will,  in  case  the  body  has  a  decided  color, 
produce  dichroism  as  well.  In  fact  Kundt,  with  the  aid  of 
Haidinger's  lens,  demonstrated  the  existence  of  this  "  temporary 
dichroism",  first  in  stretched  plates  of  india-rubber  and  gutta- 
percha. 

The  fact  that  in  crystals,  just  as  in  amorphous  bodies, 
directed  pressure  or  tension  produces  a  change  in  the  constitu- 
tion of  the  ether,  was  observed  by  Brewster  as  early  as  the  year 
1815;  yet  it  is  only  recently  that  F.  Neumann's  theory  of  the 
variation  of  the  optical  properties  of  amorphous  bodies  by 
elastic  strains  was  applied  to  crystals.  This  was  done  by  Pokels, 
who  showed  also  in  what  way  the  variation  of  the  optical  index- 
surface  is  connected  with  the  strain  ellipsoid  of  the  crystal. 

[To  explain  this  connection  it  is  necessary  to  take  into  account  the 
behavior  of  the  crystal  itself  in  respect  of  elastic  strain;  and  let  us  first  con- 
sider the  relation  of  stress  to  strain  in  the  most  simple  case.]  If  a  homogene- 
ous prism-shaped  bar  of  square  or  rectangular  cross-section  is  subjected  to  a 
compressive  or  tensive  stress  parallel  to  its  long  direction  by  means  of  a  weight 


ELASTIC  STRAIN  NOT  HOMOGENEOUS 


27I 


P  resting  on  or  hanging  from  one  end,  while  the  other  end  remains  fixed  and 
the  bar  in  vertical  position,  the  bar  suffers  a  corresponding  strain,  denoted  by 
A;  if,  therefore,  it  had  previously  the  length  L,  it  now  has  the  length  L  ±  Jl. 
Observations  made  on  a  bar  of  rectangular  cross-section  whose  breadth,  jB, 
and  whose  thickness  T  —  the  cross-section  thus  having  the  area  B T — are 
slight  as  compared  with  its  length,  show  that  so  long  as  the  elastic  limit  is 
not  transgressed  the  strain  X  is  determined  by  the  equation 

X  -   ^  •  E 
~  BT 

In  other  words,  the  lengthening  or  shortening  of  the  bar  is  proportional  to  its 
length  and  to  the  stress  producing  the  strain,  as  well  as  inversely  proportional 
to  the  cross-section  of  the  bar;  but,  in  addition,  it  is  proportional  to  E,  a  factor 
specific  to  the  substance  of  the  bar  and  independent  of  its  dimensions.  This 
quantity  is  termed  the  "coefficient  of  extension".  If  we  place  P  =  i  (i  gram, 
for  example),  L  =  i  (e.g.  i  meter),  B  =  T  =  i  (as  i  millimeter),  then  X=E\ 
that  is,  the  coefficient  of  extension  is  the  elongation  experienced  by  a  rectan- 
gular bar  of  unit  length  and  unit  cross-section  when  loaded  with  unit  weight. 

If  by  methods  depending  on  the  above  we  determine  the  coefficient  of 
extension  of  an  amorphous  body,  as  glass,  we  find  it  always  the  same,  in 
whatever  direction  a  bar  is  cut  from  the  body.  Thus,  for  a  given  amorphous 
substance  the  coefficient  has  a  constant  value  independent  of  the  direction. 
That  this  quantity  E  is  independent  of  the  direction  applies,  however,  only  to 
amorphous  bodies.  If  on  the  other  hand  we  cut  bars  in  different  directions 
from  a  crystal,  we  find  that,  although  bars  whose  long  direction  has  the  same 
crystallographic  orientation  have  the  same  coefficient  of  extension,  bars  ori- 
ented differently  from  one  another  give  different  values  for  this  quantity  ; 
that  is  to  say,  in  a  crystal  the  coefficient  of  extension  varies  with  the  direction. 

The  manner  in  which  this  coefficient  varies  with  the  direction  depends 
on  a  series  of  quantities  called  the  "constants  of  elasticity"  of  the  crys- 
tallized substance  in  question;  these  constants  number  twenty-one  in  the 
most  general  case,  while  with  the  majority  of  crystals  they  are  reduced  in 
number,  several  of  them  becoming  equal  in  accordance  with  definite  laws. 
As  for  what  concerns  more  especially  the  coefficient  of  extension,  i.e.  the  strain 
in  unit  length  along  a  direction  in  which  there  operates  unit  stress,  a  general 
view  of  its  characteristic  dependence  on  the  crystallographic  direction  is 
obtained,  if  from  a  definite  point  one  imagines  as  laid  off  in  every  direction  the 
length  proportional  to  the  quantity  E  for  the  crystallographic  direction  in 
question  and  thinks  of  the  extremities  of  all  these  lengths  as  connected  by  a 
closed  curved  surface,  —  the  so-called  "surface  of  extension  coefficients". 
This  surface  (which,  according  to  what  was  stated  above,  is  for  the  amor- 


272  INFLUENCE   OF   OTHER  PROPERTIES 

phous  bodies,  and  them  only,  a  sphere)  has  for  different  crystals  a  multi- 
farious and,  in  part,  most  complicated  form;  but  it  is  always  centrally 
symmetrical,  the  two  extremities  of  its  every  diameter  always  being  equi- 
distant from  the  center  of  the  surface.  According  to  which  of  the  elas- 
ticity constants  are  equal,  the  surface  calculated  from  the  remaining  ones 
assumes  a  more  or  less  symmetrical  form,  which  stands  in  a  regular  relation 
to  the  symmetry  of  the  geometric  form  of  the  crystals  in  question,  i.e.  to 
the  existence  in  those  crystals  of  a  larger  or  smaller  number  of  equivalent 
directions;  and  these  directions  then  prove  to  be  equivalent  in  respect 
also  of  the  elongation  parallel  to  them.  As  is  taught  by  theory  and  con- 
firmed by  experiment,  there  exist,  with  reference  to  the  form  of  the  surface  of 
extension  coefficients,  nine  different  groups,  of  crystals. 

The  first  of  these,  designated  as  Group  I,  includes  all  the  singly  refract- 
ing crystals.  For  this  group  the  surface  in  question  has  three  equal  diameters, 
which  are  mutually  perpendicular  and  correspond  either  to  the  absolute  maxi- 
mum or  to  the  absolute  minimum  of  the  extension  coefficient;  likewise  equal  are 
the  four  diameters  that  include  equal  angles  with  these  three,  the  latter  diameters 
being  the  shortest  when  the  former  are  the  longest,  and  vice  versa;  and  each 
of  the  four  planes  passing  through  the  center  perpendicular  to  the  latter  direc- 
tions intersects  the  surface  in  a  circle. 

The  optically  uniaxial  crystals  constitute  Groups  II,  Ilia,  Illb,  IVa,  IVb 
(the  "a"  and  the  ub"  groups  corresponding  to  a  higher  and  a  lower  symmetry 
of  surfaces  otherwise  similar).  The  surface  for  Group  II  is  a  rotation  figure 
produced  by  rotation,  about  the  optic  axis,  of  a  curve  symmetrical  to  that  axis 
and  having  maxima  and  minima  both  in  the  optic-axial  direction,  in  the 
direction  perpendicular  to  it,  and  between  these  two  directions.  For  Groups 
Ilia  and  Illb  the  surface  has  only  one  circular  section,  which  passes  through 
the  center  perpendicular  to  the  optic  axis.  For  IVa  and  IVb  the  surface  has 
no  longer  any  circular  section,  has  no  plane,  therefore,  in  which,  under  the 
same  circumstances,  the  elongation  is  equal  in  all  directions;  but  its  diameter 
is  still  equal  in  two  directions,  these  lying  mutually  perpendicular  in  the 
plane  normal  to  the  optic  axis. 

With  the  optically  biaxial  crystals,  comprising  Groups  V,  VI,  and  VII, 
the  surface  of  extension  coefficients  not  only  has  no  circular  section,  but  its 
symmetry,  which  for  the  preceding  groups  has  become  less  and  less,  now 
almost  entirely  vanishes.  For  Group  V,  which  includes  only  those  crystals 
whose  index-surface  has  identical  orientation  for  all  colors,  the  sur- 
face of  extension  coefficients  is  still  symmetrical  with  reference  to  three 
mutually  perpendicular  planes;  for  VI,  embracing  the  biaxial  crystals  in 
which  only  one  axis  of  the  index-surface  is  the  same  for  all  colors,  the 


ELASTIC  STRAIN  NOT  HOMOGENEOUS  273 

extension-surface  is  symmetrical  only  to  one  plane;  for  VII,  finally,  or  those 
biaxial  crystals  in  which  all  the  vibration  directions  are  dispersed,  the  surface 
in  question  can  be  symmetrical  only  to  its  center.  *  t 

Since,  then,  the  behavior  of  a  crystal  with  regard  to  com- 
pression and  extension  is  different  according  as  it  belongs  to  one 
or  another  of  the  nine  groups  mentioned  above,  there  is  a  cor- 
responding difference  in  the  variation  of  the  optical  properties. 

1.  SINGLY  REFRACTING  CRYSTALS,  subjected  to  pressure  (or 
tension)  of  whatever  crystallographic  orientation  from  one  side, 
become  doubly  refracting;  the  optical  change  is  proportional  to 
the  pressure,  and  for  any  direction  of  the  same  the  form  and 
orientation  of  the  optical  index-surface,  generally  biaxial,  may 
be  calculated,  by  means  of  the  above-mentioned  theory  (which 
Pokels  put  to  experimental  test  for  several  bodies  belonging  in 
this  class),  from  the  following  data:   First,  the  strain  existing  in 
the  crystal,  this  being  determined  by  the  elasticity  constants  of 
the  crystal  and  by  the  forces  acting  upon  it;   second,  the  optical 
constants  of  the  crystal  in  the  unaltered  state;    and  third,  cer- 
tain quantities  to  be  obtained  by  observation,  which  determine 
the  variation  of  the  optical  properties.     In  order  that  a  crystal 
of  this  group  may  become  uniaxial,  the  direction  of  the  pres- 
sure must  be  either  normal  to  one  of  the  four  circular  sections  of 
the  surface  of  extension  coefficients,  or  else  parallel  to  one  of  the 
three  mutually  perpendicular  maximum  or  minimum  diameters 
of  that  surface.     (Cf.  p.  272.) 

2.  OPTICALLY  UNIAXIAL  CRYSTALS  when  subjected  to  a  pres- 
sure   in    the   optic-axial   direction   become   either    stronger   or 
weaker  in  their  double  refraction;  but  always  remain  optically 
uniaxial.     For   then,   likewise,   the   direction   of   pressure   cor- 
responds to  a  circular  section  of  the  surface  of  extension  coeffi- 

*  [The  foregoing  analysis  is  an  abstract  of  that  to  be  found  in  Phys.  Kryst. 
4th  ed.  216-224;  for  additional  particulars  of  the  nine  groups,  and  explanatory 
figures,  see  ibid,  or  in  the  3rd  ed.  201-208.] 

f  [Group  I  comprises  the  isometric  crystal  system,  II  the  hexagonal,  III  a  and 
b  the  trigonal  (see  footnote,  p.  197),  IV  a  and  b  the  tetragonal,  V  the  orthorhombic, 
VI  the  monoclinic,  VII  the  triclinic.] 


274  INFLUENCE  OF  OTHER  PROPERTIES 

cients,  or  else  to  a  direction  perpendicular  to  which  there  exist 
two  directions  having  the  same  extension  coefficient.  (Cf.  on 
p.  272,  Groups  II-IVb.)  But  acting  in  any  other  direction, 
i.e.  from  one  side,  the  pressure  effects  a  strain  that  gives  the 
crystal  biaxial  properties. 

For  observing  the  phenomena  that  appear  in  this  case,  ser- 
vice is  made  of  the  apparatus  (pictured  in  Fig.  119)  constructed 
by  Bucking,  which  enables  one  at  the  same  time  to  measure 
the  intensity  of  the  pressure  exerted  on  the  crystal.  The  ap- 
paratus consists  in  the  first  place  of  a  brass  disk,  £>,  with  a  hole 
in  its  center,  and  this  disk  can  be  fastened  to  the  Norrenberg 
polariscope  in  place  of  the  rotatable  stage  for  carrying  the  crystal. 


Upon  the  brass  disk  there  is  screwed  firmly  a  steel  plate,  d, 
and  on  the  opposite  side  of  the  opening,  o,  in  the  disk  there 
is  a  second  steel  plate,  e,  movable  between  two  guide-bars,  /,/; 
the  crystal  plate  to  be  investigated,  resting  over  the  opening,  o, 
is  laid  against  the  first  steel  plate,  d,  and  compressed  through 
the  medium  of  the  second  plate,  e,  by  turning  the  screw,  m. 
The  latter  passes  through  a  strong  brass  frame,  r,  in  the 
other  end  of  which,  at  t,  is  drilled  a  round  hole;  movable 
lengthwise  in  this  hole  is  a  cylindrical  brass  rod,  n,  carry- 
ing at  its  end  a  disk,  <?;  and  the  sides  of  the  frame,  r,  are  let 
into  the  edge  of  q  in  such  a  way  that  the  latter  can  move  only 
longitudinally,  and  not  rotate.  About  the  cylindrical  rod,  w, 
there  is  wound  a  strong  spiral  spring,  which  presses,  when  in  the 
compressed  state,  at  its  one  end  against  the  cross-bar,  /,  of 
the  frame  and  at  its  other  end  against  the  brass  disk,  q;  at  the 
center  of  the  latter,  on  the  side  turned  away  from  the  rod,  n, 
there  is  a  pin,  which  projects  into  a  depression  in  the  plate,  d. 


ELASTIC  STRAIN  NOT  HOMOGENEOUS  275 

If,  then,  after  the  crystal  plate  has  been  inserted  the  screw,  m,  is 
turned,  the  frame,  r,  is  brought  closer  to  the  head  of  the  screw 
and  the  spiral  spring  thereby  compressed;  the  resisting  stress  in 
the  spring  acts  through  q  upon  the  plate  d,  also  through  m  upon 
the  plate  e,  and  thus  upon  the  crystal  as  well,  lying  between 
those  two  plates.  One  of  the  longer  sides  of  the  frame  is  gradu- 
ated, and  the  zero  point  of  the  scale  corresponds  to  the  position 
occupied  by  the  disk  q  when  the  spring  is  under  no  strain; 
through  this  graduation  the  compression  of  the  spring,  and 
thereby  the  pressure  exerted  on  the  crystal,  can  be  read  off 
directly  in  kilograms. 

With  the  aid  of  this  apparatus  it  is  easy  to  satisfy  one's  self 
as  to  the  presence  or  absence  of  double  refraction  in  a  cube  pre- 
pared from  glass  or  a  singly  refracting  crystal  and  subjected  to 
pressure.  To  examine  the  behavior  of  a  uniaxial  crystal  we  cut 
from  such  a  crystal  a  small  cube  having  two  of  its  pairs  of  faces 
parallel  to  the  optic  axis,  the  third  pair  perpendicular  to  the  axis. 
If  we  bring  this  cube  between  the  two  steel  plates  d  and  e  in  such 
a  way  that  two  faces  of  the  former  kind  lie  against  the  plates, 
the  optic  axis  of  the  crystal  thus  standing  perpendicular  to  the 
plane  of  the  figure,  then  on  our  turning  the  screw  the  crystal 
undergoes  a  compression  whose  direction  is  perpendicular  to  the 
optic  axis.  If,  then,  the  compression  apparatus  is  joined  to  the 
polariscope  (for  convergent  light),  one  beholds  in  the  instru- 
ment, so  long  as  the  plate  is  subjected  to  no  pressure,  the 
normal  interference-figure  of  the  uniaxial  crystal.  But  if  one  turns 
the  screw,  the  circular  color  rings  begin  to  elongate  into  ellipses; 
and  when  the  direction  of  this  elongation  lies  diagonal  to  the 
principal  sections  of  the  nicols  of  the  instrument,  the  arms  of 
the  cross  separate  at  the  center  and  become  transformed  into 
hyperbolas.  In  short,  there  arises  the  interference-figure  of  a 
biaxial  crystal  with  small  axial  angle,  the  size  of  this  angle 
increasing,  however,  with  the  pressure.  As  regards  the  plane 
of  the  two  optic  axes  now  existent,  its  direction  depends  on 
whether  the  double  refraction  of  the  uniaxial  crystal  in  question 


276  INFLUENCE   OF  OTHER   PROPERTIES 

was  positive  or  negative.  If,  as  in  glass,  (see  p.  264)  by 
the  pressure  the  velocity  of  light  rays  vibrating  in  the  direction 
of  pressure  is  increased,  then  the  result  as  regards  the  position 
of  the  axial  plane  must,  with  the  two  kinds  of  uniaxial  crystals, 
be  exactly  the  opposite,  —  as  is  readily  seen.  For,  since  with 
positive  crystals  the  rays  vibrating  parallel  to  the  axis  have 
the  least  velocity  and  those  vibrating  perpendicular  to  it  the 
greatest,  then,  if  for  the  latter  rays  the  velocity  is  by  the  pres- 
sure increased,  the  vibration  direction  of  these  rays  is  still  that 
of  the  greatest  light  velocity,  and  the  vibration  direction  lying 
perpendicular  to  this  direction  in  the  same  plane  becomes  that 
of  the  intermediate  light  velocity,  while  the  direction  that  was 
the  optic  axis  remains  the  vibration  direction  of  the  least  light 
velocity;  and  since  the  optic  axes  of  a  biaxial  body  always  lie  in 
the  plane  of  the  first-  and  the  third-named  of  these  three  vibra- 
tion directions,  the  mutual  recession  of  the  two  axes  in  conse- 
quence of  the  pressure  occurs  in  a  plane  that  is  parallel  to  the 
direction  of  pressure.  Since  the  axial  angle  resulting  from  pres- 
sure is  always  small,  what  was  previously  the  optic  axis  always 
remains  the  acute  bisectrix;  and  thus  the  biaxial  crystal  arising 
from  a  positive  uniaxial  one  is  likewise  positive.  In  the  case  of 
the  negative  crystals  the  plane  perpendicular  to  the  optic  axis 
contains  all  the  vibration  directions  of  the  least  light  velocity; 
this  light  velocity  is  increased  in  the  direction  of  pressure,  in 
consequence  whereof  the  direction  named  becomes  vibration 
direction  of  the  now  intermediate  light  velocity  and  the  vibration 
direction  lying  perpendicular  to  it  in  the  same  plane  that  of  the 
least,  while  the  direction  that  was  previously  the  optic  axis 
remains  vibration  direction  of  the  greatest  light  velocity;  accord- 
ingly, the  mutual  recession  of  the  optic  axes  takes  place  in  a  plane 
standing  perpendicular  to  the  direction  of  pressure,  and  the  arising 
biaxial  crystal  is  negative.  Now  the  above-described  behavior 
is  exhibited  in  fact  by  all  the  uniaxial  crystals  investigated  up 
to  this  time;  moreover,  Bucking's  experiments  have  shown  that 
in  a  uniaxial  crystal  a  relatively  slight  pressure  suffices  to  call 


ELASTIC   STRAIN  NOT  HOMOGENEOUS  277 

forth  a  small  axial  angle,  but  that  in  order  to  make  the  angle 
larger  a  very  much  greater  pressure  is  requisite,  —  proving  that 
the  variation  of  the  axial  angle  is  not  proportional  to  the  pressure. 
Particularly  interesting  results  are  yielded  by  the  applica- 
tion of  pressure  to  such  uniaxial  crystals  as  rotate  the  polari- 
zation plane  of  light;  e.g.  to  quartz.  This  crystal  is  positive  in 
its  double  refraction  and  therefore  acquires,  when  compressed 
at  right  angles  to  the  axis,  two  axes  in  the  plane  parallel  to 
the  direction  of  pressure.  In  the  arising  axial  figure,  at  the 
centers  of  the  rings  surrounding  the  two  axes,  a  coloring  then 
appears  which  on  rotation  of  the  analyzer  varies,  —  just  as  such 
a  coloring  appears  at  the  center  of  the  interference-figure  of  an 
uncompressed  quartz  plate  (see  p.  230) ;  so  it  follows  that  in  the 
two  optic-axial  directions  as  well  there  must  occur  a  rotation  of 
the  polarization  plane  of  light.  The  experiments  conducted  by 
Mach  and  Merten  have  shown,  first,  that  in  each  of  these  direc- 
tions there  are  transmitted,  not  two  circularly  vibrating  rays, 
but  two  rays  vibrating  in  ellipses;  second,  that  in  each  direction 
the  two  vibrations  take  place  in  the  opposite  sense  and  on 
emerging  combine  to  form  one  single  vibration,  whose  path  is 
also  an  ellipse;  third,  that  the  major  axis  of  this  elliptical  path 
is  rotated  in  the  sense  of  the  ray  having  the  greater  transmission 
velocity;  fourth  and  finally,  that  therewith  the  difference  of 
path,  as  compared  with  that  of  the  two  circular  rays  in  normal 
quartz,  has,  by  the  pressure,  even  been  augmented.  Taking 
into  account  that,  according  to  Gouy's  theory  (see  p.  223),  the 
path  difference  produced  by  the  ordinary  double  refraction  is 
superposed  on  that  called  forth  by  the  circular,  Beaulard  carried 
out  a  research  by  investigating  the  ellipticity  of  light  rays  that 
after  having  been  plane-polarized  were  caused  to  pass  through 
compressed  quartz  in  various  directions;  his  conclusions  are 
that  by  pressure  the  rotatory  power  of  quartz  suffers  no  change, 
only  the  double  refraction  being  altered,  and  that  in  each  of  the 
two  optic-axial  directions  (forming  in  the  case  in  question  an 
angle  of  13°)  there  exists  normal  circular  double  refraction. 


278  INFLUENCE  OF   OTHER   PROPERTIES 

3.  OPTICALLY  BIAXIAL  CRYSTALS,  subjected  to  pressure  par- 
allel to  one  of  their  three  principal  vibration  directions,  must 
experience  a  variation  of  their  axial  angle,  since  the  pressure 
effects  a  change  in  the  ratios  among  the  axial  lengths  of  the 
optical  index-surface,  and  on  these  ratios  depends  the  size  of 
that  angle.  The  biaxial  crystals  as  yet  investigated  with 
Bucking's  appliance  have  shown  that  with  them,  too,  the  pres- 
sure effects  an  increase  in  the  velocity  of  rays  vibrating  in  the 
direction  of  pressure.  Suppose,  for  example,  that  we  have  a 
cube  cut  from  a  negative  crystal,  with  its  sides  parallel  to  the 
three  principal  optic  sections,  and  let  this  cube  be  so  placed  in 
the  instrument  that  the  acute  bisectrix  stands  vertical,  —  so 
placed  therefore  that  in  the  instrument  we  see  the  interference- 
figure  with  the  lemniscates.  Then,  if  the  crystal  is  compressed 
in  the  direction  of  the  F-axis  of  the  index-surface,  i.e.  perpen- 
dicular to  the  axial  plane,  the  intermediate  light  velocity  be- 
comes greater;  it  thus  approaches  the  greatest  light  velocity,  and, 
since  this  latter  is  the  velocity  parallel  to  the  acute  bisectrix, 
the  axial  angle  must  increase.  Now  if  we  rotate  the  crystal  90° 
about  the  acute  bisectrix  and  subject  it  to  pressure  from  the 
other  two  sides,  i.e.  parallel  to  the  vibration  direction  of  least 
light  velocity,  this  light  velocity  must  in  its  turn  become  greater, 
thus  approaching  the  intermediate;  in  consequence  the  lengths 
of  the  two  axes  of  the  index-surface  that,  standing  perpendicular 
to  the  acute  bisectrix,  lie  horizontal  in  the  instrument  are  then 
less  different  from  each  other,  and  the  crystal,  in  its  optical 
properties,  approaches  one  in  which  these  two  directions  are 
equivalent;  in  other  words,  it  grows  more  like  a  uniaxial  crystal, 
and  thus  the  axial  angle  must  become  smaller.  It  is  plain  that 
in  the  case  of  a  small  axial  angle,  i.e.  when  the  principal  refrac- 
tive indices  ft  and  f  differ  but  little  even  at  the  outset,  a  certain 
pressure  will  suffice  to  make  the  two  quite  equal;  the  crystal 
then  becomes  negatively  uniaxial.  If  the  pressure  is  brought 
still  higher,  then  what  previously  was  7-  becomes  smaller  than  /?, 
the  Z-axis  of  the  index-surface  thus  becoming  the  intermediate 


OPTICALLY  ANOMALOUS   CRYSTALS  279 

axis;  so  the  optic  axes  must  recede  from  each  other,  but  in  a 
plane  standing  perpendicular  to  their  former  plane.  That  the 
crystal  is  uniaxial  under  the  pressure  with  which  /?  =  7  applies, 
however,  only  to  one  single  color,  since  the  change  that  pressure 
produces  in  the  transmission  of  light  is  unequal  for  different 
colors;  therefore,  if  we  employ  a  pressure  that  makes  the  crys- 
tal uniaxial  for  medium  colors,  the  optic  axes  for  the  one  part 
of  the  spectrum  still  lie  in  the  same  plane  as  before,  while  for  the 
other  part  they  have  already  spread  out  in  the  new  axial  plane; 
such  a  plate  accordingly  behaves,  when  its  dispersion  is  great 
enough,  like  the  substances  having  crossed  axial  planes,  de- 
scribed on  pages  172-173.  With  positive  crystals,  as  is  taught  by 
an  analogous  consideration,  a  pressure  acting  perpendicular  to  the 
plane  of  the  axes  will  diminish  their  angle  (transforming  the 
crystal,  when  it  has  only  a  small  angle  to  start  with,  into  one 
that  is  positively  uniaxial),  while  a  pressure  applied  parallel  to 
the  axial  plane  will  make  the  axial  angle  larger.  It  is  evident, 
finally,  that,  when  the  direction  of  pressure  is  parallel  to  none  of 
the  three  principal  vibration  directions,  but  oriented  in  any 
other  way,  there  must  occur  a  variation  not  only  in  the  form, 
but  also  in  the  crystallographic  orientation,  of  the  optical 
index-surface. 

Optically  Anomalous  Crystals 

We  have  learned  from  the  foregoing  that  definitely  directed 
pressure  or  tension  in  a  crystal  has,  as  its  effect,  that  the  crystal 
takes  on  properties  different  from  those  pertaining  regularly  to 
the  group  of  crystals  to  which  the  crystal  in  question  normally 
belongs.  When  the  forces  in  question  do  not  act  uniformly  on 
all  parts  of  the  crystal,  the  arising  phenomena  as  well,  just  as 
in  the  strained  amorphous  bodies  described  on  page  264  et  seq.t 
vary  with  the  locality  in  the  crystal;  the  crystal,  too,  has  then 
ceased  to  be  homogeneous.  Now  supposing  that,  in  a  crystal, 
from  any  cause  the  deviations  of  its  optical  properties  from  those 
in  the  normal  state  of  the  crystal  have  become  permanent, 


280  INFLUENCE  OF  OTHER   PROPERTIES 

the  crystal  is  said  to  be  optically  anomalous.  (Cf.  p.  219.)  In 
the  diamond,  for  example,  a  singly  refracting  crystal,  by  reason 
of  inclusion  during  the  formation  of  the  crystal  there  not  in- 
frequently exist  foreign  bodies  (crystals  of  other  minerals),  and 
these  are  surrounded  by  a  zone  in  which  the  diamond  exhibits 
distinct  double  refraction.  The  cause  of  this  phenomenon  is 
the  inequality  between  the  coefficients  of  expansion  of  the  two 
bodies,  whose  crystallization  undoubtedly  took  place  at  a  very 
high  temperature:  in  consequence,  while  cooling  to  the  present 
temperature  the  two  bodies  experienced  a  different  degree  of 
contraction;  so  that  now  the  parts  where  they  border  on  each 
other,  whatever  be  the  direction  of  the  normal  to  the  boundary- 
surface,  must  be  in  a  state  of  compression  or  of  extension.  In 
colored  crystals,  since  in  colored  bodies  pressure  influences 
also  the  absorption  of  light  (see  p.  2 70), 'when  the  crystal  in- 
cludes smaller  ones  of  another  substance  there  may  arise  also 
pleochroic  halos,  in  the  manner  described. 

Crystals  having  great  plasticity  (see  p.  283),  which  accord- 
ingly very  easily  experience  permanent  strains,  frequently  suffer 
local  compressions,  and  in  consequence  there  arise  optical 
anomalies.  Thus  the  singly  refracting  halite  (rock-salt)  ex- 
hibits after  compression,  or  indeed  even  after  breaking  it 
roughly,  the  stripes  characteristic  of  double  refraction. 

Many  other  crystals  that,  to  judge  from  their  other  proper- 
ties, belong  like  those  mentioned  above  to  the  singly  refracting 
group,  but  which  contain  no  recognizable  inclusions  and  have 
been  subjected  to  no  strain  subsequent  to  their  formation,  ex- 
hibit nevertheless  throughout  their  whole  extent  distinct  phenom- 
ena of  double  refraction;  these  phenomena  are  such  that  the 
crystals  appear  to  be  composed  of  regularly  arranged  parts  in 
which  the  vibration  directions  are  differently  oriented.  There 
are  likewise  crystals  that,  although  optically  uniaxial  in  form, 
are  really  biaxial,  having  moreover  in  different  parts  axial  angles 
of  different  size  and  axial  planes  of  different  orientation.  These 
optical  anomalies  (which,  like  the  amorphous  bodies  described 


OPTICALLY  ANOMALOUS  CRYSTALS  28i 

on  page  264  et  seq.,  were  observed  first  by  Brewster),  Reusch 
sought  to  explain  by  the  assumption  that  during  their  crystalli- 
zation the  substances  in  question,  similarly  to  solidifying  colloidal 
bodies,  experience  a  certain  contraction,  and  that  in  conse- 
quence there  arise  in  the  crystal  a  permanent  strain.  In  crystals 
of  alum,  for  example,  the  double  refraction  looks  as  though  the 
substance  of  the  crystal  were  strained  within  certain  planes  and 
as  though,  while  the  crystal  was  being  built  up,  the  superposition 
in  layers  had  taken  place  parallel  to  these  planes.  The  distri- 
bution of  the  anomalous  state  in  question  would  then  of  neces- 
sity stand  in  a  certain  relationship  to  the  outward  form  of  the 
crystal;  and  such  is  indeed  the  case  with  substances  of  the  kind 
now  considered.  For  this  reason,  in  connection  with  the  above 
explanation  of  the  phenomenon,  the  experiments  are  of  interest 
that  have  been  carried  on  with  amorphous  bodies  to  which  a 
shape  similar  to  that  of  the  crystals  in  question  was  given  arti- 
ficially. To  this  end  Ben  Saude  prepared  molds  of  crystal 
models  and  filled  them  with  gelatin,  allowing  the  latter  to  dry 
two  or  three  days  before  removal  from  the  mold;  he  then  cut 
plates  from  the  gelatin  in  definite  directions,  and  in  order  to 
prevent  further  drying  out  the  plates  were  packed  in  Canada 
balsam.  Now,  as  a  matter  of  fact,  these  gelatin  plates  did  in- 
deed exhibit  double-refraction  phenomena  analogous  to  those 
that  are  exhibited  by  optically  anomalous  crystals  having  the 
same  shape  as  the  gelatin  models;  in  particular,  they  mani- 
fested a  division  into  sectors  extinguishing  in  different  directions, 
the  position  of  the  sector  boundaries  depending  on  the  shape  of 
the  model. 

But  there  is  no  way  of  proving  Reusch's  assumption — that 
during  the  process  of  crystallization  a  crystallized  substance, 
like  a  colloid  passing  gradually  over  into  the  solid  state,  suffers  a 
change  in  volume.  Accordingly,  as  an  explanation  opposed  to 
the  foregoing,  it  was  suggested  by  Mallard  that  the  crystals  in 
question  might  be  built  up  of  interpenetrating  parts,  these  being 
very  thin  lamellae  of  the  crystal  substance,  normally  constituted 


282  INFLUENCE  OF  OTHER  PROPERTIES 

but  differently  oriented.  That  the  crystals  must  then  exhibit  the 
phenomena  mentioned  has  already  been  set  forth  on  pages  217- 
219.  And  in  fact,  for  very  many  crystals  that  formerly  were 
termed  optically  anomalous,  Mallard  has  demonstrated  the  cor- 
rectness of  this  explanation.  Since,  then,  with  such  composite 
masses,  which  in  their  optical  relations  behave  differently  at 
different  points,  it  is  a  matter  of  crystals  that  are  only  appar- 
ently incomposite,  the  constitution  of  their  differently  oriented 
parts  being  optically  normal,  we  may  not  properly  speaking 
designate  such  a  mass  as  an  optically  anomalous  crystal,  as  is 
often  done.  The  cases  of  true  optical  anomaly  in  a  crystal 
(leaving  out  of  the  question  the  temporary  state  described  on 
pp.  273-279)  seem  to  reduce  themselves  to  (i)  those  produced 
by  inclusions  (see  p.  280)  and,  possibly,  to  (2)  certain  phe- 
nomena exhibited  by  so-called  "  isomorphous  mixtures".  (See 
p.  6.)  In  the  bodies  last-named,  which  are  crystals  built  up  of 
the  smallest  particles  of  two  salts,  the  dimensions  of  the  two 
kinds  of  component  particles,  although  very  similar,  are  not 
exactly  the  same;  and  accordingly  it  seems  very  possible  that 
owing  to  this  lack  of  molecular  homogeneity  there  may  arise  per- 
manent strains  of  the  whole  mass.  This  view  is  supported  by 
the  observations  of  Klocke  and  Brauns:  according  to  these 
observations  crystals  of  alum  exhibit  normal  single  refraction 
when  they  consist  of  the  chemically  pure  salt,  but  are  dis- 
tinctly birefringent  when  they  contain  isomorphous  admixtures; 
so  that  here  we  probably  have  to  do  with  a  true  optical 
anomaly. 

ELASTIC  STRAIN  BY  ELECTRICAL  ACTION 

[Elastic  strain  in  crystals,  and  consequent  variation  of  the  opti- 
cal properties,  may  be  induced  in  still  another  way  than  by  thermic 
expansion  and  by  the  action  of  mechanical  forces  on  the  crystal.] 

Certain  crystals  that  are  non-conductors  of  electricity  exhibit  in  definite 
mutually  opposite  directions  a  difference  in  their  behavior,  in  that,  when  they 
are  heated  and  then  allowed  to  cool,  they  assume  electric  polarity,  the  polarity 


PLASTICITY  283 

is  developed  only  on  variation  of  the  temperature:  it  disappears  when  the 
latter  has  become  stationary.  This  property  is  termed  "  polar  pyroelectricity  ", 
and  the  directions  in  which  the  polarity  is  excited  we  call  "electric  axes". 
Since  compression  of  a  body  operates  like  lowering  its  temperature,  and  ex- 
tension like  raising  the  temperature,  it  is  to  be  expected  that  in  polar  pyro- 
electric  crystals  electropolarity  would  be  excited  in  the  electric-axial  directions 
during  compression  or  extension  as  well;  and  the  electricity  produced  at  one 
of  the  poles  of  such  an  axis  by  the  diminution  of  pressure  would  necessarily 
be  the  opposite  of  that  called  forth  at  the  same  pole  by  the  increase.  This 
property  of  polar  pyroelectric  crystals  bears  the  name  "  polar  piezo-electricity". 

Just  as  electricity  is  produced  by  elastic  deformation  of  a 
polar  dielectric  crystal,  so  too  does  electrization  of  such  a  body 
effect  a  deformation  of  the  body.  While  J.  and  P.  Curie  proved 
this  by  direct  measurement,  Kundt  showed  that  the  circular 
color  rings  exhibited  by  a  quartz  plate  in  convergent  polarized 
light  pass  over,  when  viewed  in  an  electrostatic  field,  into  ellipses 
such  as  arise  in  consequence  of  pressure  directed  perpendicular 
to  the  optic  axis.* 

PERMANENT  STRAIN 
Plasticity 

If  a  straining  force  acting  on  a  crystal  exceeds  the  elastic 
limit  of  the  crystal,  the  latter  suffers  deformations  that  are 
lasting  and  generally  unhomogeneous,  [so  that  permanent  vari- 
ations in  the  optical  properties  must  arise.  (Cf.  pp.  280  and 
285.)]  The  property  of  a  body  to  suffer,  when  subjected  to 
external  stresses  exceeding  its  elastic  limit,  changes  in  shape 
which  are  permanent  but  also  gradual  and  steady  (without 
losing,  through  any  sudden  occurrence,  the  connection  of  its 
parts),  is  called  "plasticity".  Many  crystals  are  so  plastic  that 
the  slightest  forces  suffice  to  change  their  shape,  and  with  some 
the  elastic  limit  is  so  nearly  zero  that  they  behave  like  a  liquid. 

*  [According  to  Pokels  (F.  Pokels,  Abh.  Ges.  d.  Wissensch.  Gottingen,  39, 1-204) 
the  change  is  too  great  to  be  a  consequence  of  the  deformation  alone,  but  must  be 
due  in  part  to  a  direct  influence  of  the  electric  force  on  the  light- vibrations  within 
these  crystals.] 


284  INFLUENCE   OF   OTHER  PROPERTIES 

According  to  Lehmann's  researches  an  aggregate  of  such 
crystals  —  "  flowing  crystals ",  as  it  were  —  can  be  brought 
by  pressure  or  tension  into  parallel  orientation  (Lehmann's 
"  homeotropy ");  and  if  the  individual  crystals  are  doubly 
refracting,  the  mass  of  course  then  exhibits  parallel  extinction. 
Furthermore,  the  surface  tension  can  even  exceed  the  elastic 
limit,  and  then  each  crystal  assumes  the  form  of  a  drop,  just  like 
a  perfect  liquid,  whose  elastic  limit  is  zero  (Lehmann's  "  liquid 
crystals").* 

Gliding 

In  addition  to  cleavage  planes  there  exist  in  crystals  planes 
distinguished  by  the  fact  that  parallel  to  them,  in  a  definite  direc- 
tion (but  not  necessarily  in  the  opposite  direction  as  well),  there 
can  take  place  with  particular  ease  a  slipping,  or  "  gliding",  of 
the  crystal  particles  on  one  another.  The  result  is  that,  when 
for  example  the  crystal  is  pressed,  its  particles  become  displaced 
relatively  to  one  another  along  such  a  plane,  and  sometimes 
even  completely  separate. 

By  Reusch,  who  was  the  first  to  point  out  their  existence,  these  planes 
were  called  "gliding-planes"  (Gleitflachen).  In  certain  crystals  they  appear 
as  planes  of  easy  separation  ("parting"),  if  the  point  of  a  steel  cone  (the 
center-punch  used  by  metal-workers  is  particularly  well  adapted  for  this  pur- 
pose) is  set  on  the  surface  of  the  crystal  and  driven  in  by  a  sharp  tap  with 
a  light  hammer.  The  so-called  "percussion -figure",  obtained  by  this  method, 
the  "percussion-test",  consists  of  rectilinear  cracks  radiating  from  the  point 
of  percussion  in  one  or  more  directions;  these  cracks  usually  correspond  to 
the  gliding-planes,  not  to  the  cleavage  planes,  —  as,  for  example,  in  crystals 
of  halite.  These  crystals  belong  in  the  singly  refracting  class  and  therefore 
(see  p.  272)  possess  three  mutually  perpendicular  directions  in  which  they 
behave  alike  with  respect  to  the  influence  of  external  forces;  parallel  to  these 
directions  their  strength  is  minimum,  so  that  along  planes  perpendicular  to 
these  three  directions  they  split  very  easily.  Now  on  such  a  cleavage-face  the 
percussion-figure  appears  as  a  four-rayed  star,  whose  branches  do  not  pass  out 

*  O.  Lehmann:  "Fliissige  Krystalle,  sowie  Plasticitat  von  Krystallen  in 
allgemeinen,  molekulare  Umlagerungen  und  Aggregatzustandsanderungen".  With 
483  figures  in  the  text  and  39  colored  plates.  Leipzig,  1904. 


GLIDING 


285 


parallel  to  the  two  other  cleavage  planes,  but  bisect  their  angle.  Since  all  three 
of  the  cleavage  planes  behave  alike  in  this  respect,  there  must  exist  in  the  halite 
six  gliding-planes,  which  bisect  the  angles  of  a  cube  split  from  the  crystal. 
If  a  prism  bounded  by  the  six  cube  faces  is  pressed  together  from  the  end 
faces  and  therefore  in  the  direction  of  one  of  the  edges,  it  becomes  shorter 
and  thicker  in  consequence  of  the  particles  slipping  along  the  gliding-planes 
that  form  45°  with  the  direction  of  pressure;  thus  considerable  deformations 
can  be  produced  without  fracturing  the  prism.  In  this  case,  just  as  when  the 
crystal  is  broken,  cut,  etc.,  there  always  arise  at  the  same  time,  owing  to  the 
plasticity  of  halite,  local  compressions,  which  are  to  be  recognized  by  double 
refraction  (brightening  between  crossed  nicols).  (See  p.  280.)  In  fact,  halite 
that  is  entirely  homogeneous  and  free  from  doubly  refracting  areas  is  seldom 
found. 

By  far  the  most  interesting  and  theoretically  the  most  im- 
portant phenomena  of  gliding  are  presented  by  calcite,  in  which 


I 


Fig.  120  a. 


Fig.  I20&. 


mineral,  likewise,  the  existence  of  gliding-planes  was  discovered 
by  Reusch.  On  a  cleavage-rhombohedron  of  calcite  whose 
principal  section  is  abed  in  Fig.  120  a  (ab  and  cd  being  two 
opposite  obtuse  edges,  be  and  a d  the  diagonals  of  the  end  faces) , 
let  us  imagine  two  horizontal  planes  to  be  cut  as  indicated  by 
the  dotted  lines.  Then,  if  pressure  is  exerted  on  the  calcite  in 
the  direction  of  the  arrows,  the  right  half  slips  downward  and  the 
left  half  upward.  This  movement  proceeds  in  such  a  way  that 


286  INFLUENCE   OF   OTHER  PROPERTIES 

within  the  space  between  two  planes  parallel  to  the  gliding- 
plane,  which  stand  perpendicular  to  the  plane  of  the  figure  and 
would  therefore  truncate  the  edges  ab  and  cd  evenly  before  and 
behind  this  plane,  there  takes  place  a  rearrangement  of  the  sub- 
stance of  the  crystal  into  the  symmetrically  opposite  position. 
The  arising  lamella  (see  Fig.  1206)  appears  only  on  those  two 
sides  of  the  cleavage-rhombohedron  that,  in  Fig's  120  a  and  b, 
are  represented  as  the  upper  and  the  lower  face,  parallel  to  each 
other;  it  is  seen  as  a  very  narrow,  rectilinear  stria,  which  runs 
exactly  parallel  to  the  longer  diagonal  of  the  rhombohedron  face. 
Now  it  usually  happens,  in  a  rhomb  of  this  kind,  that  a  con- 
siderable number  of  such  lamellae  arise,  so  that  on  two  of  its 
opposite  faces  the  calcite  is  seen  to  be  striated  parallel  to  the 
longer  diagonal.* 

If,  then,  a  calcite  plate  cut  perpendicular  to  the  optic  axis 
is  subjected  to  lateral  pressure,  it  becomes  biaxial.  (Axial 
plane  perpendicular  to  the  direction  of  pressure  — cf.  p.  276.) 
But  on  increasing  the  pressure,  permanent  changes  arise  in  the 
calcite.  These  changes  are  revealed  by  the  interference-figure 
in  convergent  light,  which  differs  essentially  from  the  normal: 
the  color  rings  are  much  narrower;  the  form  of  the  cross 
is  deranged;  four  dark  spots  appear  in  diagonal  position, 
connected  by  colored  arcs;  and  so  forth.  The  cause  of  this 
behavior  is  that  lamellae  arise  in  the  plate  in  the  manner 
described. 

Now  all  the  phenomena  hitherto  described,  especially  the 
striation  parallel  to  the  longer  diagonal  of  the  rhombohedron 
faces,  are  very  often  exhibited  by  calcite  in  the  natural  state, 
under  the  conditions  of  its  occurrence.  This  leads  to  the  as- 
sumption that  here,  too,  a  pressure  exerted  on  the  mineral  has 
found  relief  in  the  formation  of  the  lamellae  in  question. 

*'The  cause  of  the  hollow  passages  within  calcite  crystals,  which  are  often 
met  with  in  nature  and  which  can  also  be  produced  artificially,  has  been  traced 
to  these  lamellae  —  to  their  changing  suddenly  from  one  plane  to  another,  ending 
abruptly  within  the  crystal,  or  intersecting  one  another. 


GLIDING 


287 


While  according  to  Reusch's  procedure  there  arise  on  press- 
ing a  calcite  rhombohedron  only  a  number  of  thin  lamellae,* 
Baumhauer  has  given  us  a  method  by  which  any  part  of  a 
rhombohedron  can  be  brought  over  into  the 
symmetrically  opposite  position.  Suppose  a 
prism-shaped  cleavage-fragment  (Fig.  121) 
to  be  fixed  (best  in  a  groove  cut  to  corre- 
spond in  a  wooden  board)  with  one  of  its 
obtuse  edges  horizontal,  and  so  that  the  long  Fi 

diagonal,  ce,  of  its  right  end  face,  cdef,  is 
likewise  horizontal.  Then,  if  at  the  point,  a,  the  cutting-edge  of 
a  knife  is  set  at  right  angles  against  the  upper  edge  of  the  cal- 
cite and  gradually  pressed  in,  the  part  of  the  crystal  that  lies  to 
the  right  of  the  knife  becomes  shifted  in  such  a  way  that,  when 
the  cutting-edge  of  the  knife  has  penetrated  to  the  intermediate 
edges  of  the  calcite  (with  large  pieces  this  requires  considerable 
pressure),  it  has  assumed  the  exact  form  of  a  half-rhombohedron 
lying  in  the  opposite  position;  i.e.  the  form  of  the  mirror-image, 
with  respect  to  the  gliding-plane  through  ce,  of  the  lower  half- 
rhombohedron.  The  triangle  cef  has  now  taken  the  position  ceg\ 
the  obtuse  solid  angle/,  the  so-called  "pole-angle"  (a  in  Fig.  48, 
p.  91), —  at  the  junction  of  three  similar,  obtuse  edges, — has 
become  a  "lateral  angle",  i.e.  the  meeting-point  of  two  acute 
edges,  eg  and  eg,  and  one  obtuse  edge,  /#;  consequently  the  left 
upper  angle  of  the  rearranged  portion  has  become  a  pole- 
angle. 

Now  if  we  investigate  the  part  of  the  crystal  in  the  new 
position  more  closely,  it  proves  to  be,  physically,  entirely  homo- 
geneous; its  properties  no  longer  correspond  to  its  previous 
form,  however,  but  to  its  present  form.  For  example,  it  is 
optically  uniaxial  uniformly  at  all  points,  and  with  the  same 

*  It  is  sometimes  possible,  but  only  under  very  special  circumstances,  to 
apply  the  pressure  on  a  rhombohedron  like  the  one  in  Fig.  120  in  such  a  way  that 
the  crystal  substance  is  all  rearranged;  in  other  words,  so  that  the  obtuse  angles 
now  lie  at  b  and  d,  while  the  acute  are  at  a  and  c. 


288  INFLUENCE   OF   OTHER  PROPERTIES 

birefringence  as  before;  but  the  optic  axis,  lying  in  the  vertical 
principal  section,  does  not  incline  from  the  right  above  to  the 
left  below,  as  in  the  part  to  the  left  of  a  and  also  in  the  lower 
part  (cf.,  for  example,  Fig.  53  on  p.  97,  supposing  the  edge  ac  to 
be  horizontal) :  it  is  now  inclined,  at  the  same  angle,  from  the 
left  above  to  the  right  below,  in  correspondence  to  the  new 
position  of  the  pole-angle.*  f 

TWINNING 

For  the  normal  growth  of  a  crystal,  i.e.  in  order  that  a  crystal 
may  be  homogeneous  at  all  points,  it  is  requisite  that  in  being 
deposited  the  particles  drawn  to  it  from  the  mother-liquor  shall 
be  given,  with  respect  to  the  particles  of  the  previous  layer,  the 
same  orientation  as  these  were  with  respect  to  the  next  preced- 
ing. But  in  addition  to  this  corresponding  orientation  of  suc- 
cessive layers,  there  can  exist  a  second  orientation  in  which 
stable  equilibrium  exists  among  the  forces  acting  among  adja- 

*  This  rotation  of  the  optic  axis  amounts  to  52!°.  Since  the  entire  inner 
constitution  of  the  crystal  with  reference  to  the  new  axis  of  the  rhombohedron  has 
become  the  same  as  it  was  previously  with  reference  to  the  old  axis,  the  smallest 
particles  of  the  calcite  must  have  experienced  (in  addition  to  the  slipping)  a  rota- 
tion through  this  same  angle.  The  rotation  of  the  face  cef  (Fig.  121)  into  its  new 
position  ceg  is  considerably  less  —  only  38°. 

f  Since  calcite  shifted  along  the  gliding-plane  has  exactly  the  same  properties 
as  originally,  although  in  different  orientation,  the  arrangement  and  direction  of 
its  smallest  parts  must  correspond  in  both  cases  to  the  same  equilibrium  of  the 
internal  forces.  When,  therefore,  by  the  action  of  external  forces  such  a  shift- 
ing of  the  crytsal  substance  is  effected,  the  accompanying  rotation  of  the  par- 
ticles (see  footnote*)  must  be  such  that,  when  the  shifted  part  is  exactly 
midway  between  its  original  and  its  final  position,  the  smallest  particles  too 
have  completed  exactly  half  their  rotation.  Consequently  these  particles  can  be 
carried  over,  by  the  internal  forces  of  the  crystal,  into  the  one  position  of  equi- 
librium just  as  easily  as  into  the  other.  In  agreement  with  this  (and  as  was  found 
by  Reusch)  it  may  indeed  be  observed,  when  gliding-lamellae  are  being  produced  in 
calcite,  that  a  rearrangement  which  has  proceeded  less  than  half-way  disappears 
completely  when  the  pressure  ceases  to  act;  while  when  the  labile  state  arising 
midway  in  the  shifting  has  once  been  passed  over,  the  lamella  forms  of  itself, 
because,  owing  to  the  internal  forces,  the  particles  strive  toward  the  second  position 
of  equilibrium. 


TWINNING  289 

cent  particles.*  Under  such  conditions,  the  particles  attracted 
to  the  growing  crystal  being  of  course  present  in  the  solution  in 
all  possible  positions,  it  will  happen  just  as  often  that  a  particle 
has  nearly  the  orientation  corresponding  to  the  first  position  of 
equilibrium  as  to  the  second.  Since  the  particles  must  assume 
that  one  of  the  two  orientations  to  attain  which  the  less  work 
(i.e.  the  less  rotation)  is  necessary,  they  will  be  deposited  just  as 
often  in  the  one  orientation  as  in  the  other,  and  will  now,  in 
their  turn  likewise,  become  the  cause  of  analogous  deposition  of 
additional  particles.  Thus,  beginning  immediately  after  the 
crystallization  has  begun,  two  distinct  arrangements  of  the  par- 
ticles, differently  oriented  but  otherwise  similar,  may  grow  on 
in  the  solution;  and  the  result  is  a  so-called  "  twin  "  crystal, 
whose  two  parts  are  in  general  bounded  by  like  faces,  since  the 
conditions  of  their  crystallization  are  essentially  the  same. 

If  twinning  is  repeated  in  such  a  way  that  with  the  second 
crystal  there  combines  a  third,  after  the  same  law,  with  this 
latter  crystal  a  fourth,  and  so  on,  the  resulting  growth  is  termed 
a  "  trilling  ",  a  "  fourling  ",  etc.  According  to  the  law  of  the 
twinning,  two  cases  may  here  arise:  either  the  third  crystal  has 
a  different  position  from  the  first,  the  number  of  the  regularly 
combined  crystals  then  naturally  being  limited;  or  else  the 
third  crystal  is  parallel  to  the  first,  the  fourth  crystal,  com- 
bined with  this  third  one,  parallel  to  the  second,  and  so  forth. 
In  the  latter  case,  we  have  a  fourling  of  crystals  succeeding  one 
another  in  alternate  orientation  with  reference  to  the  common 
twinning-plane.  In  such  combinations  the  phenomenon,  quite 
common  in  twin  crystals,  of  especially  marked  development  of 
the  single  crystals  along  the  twinning-plane,  appears  in  so  high 
a  degree  that  the  single  crystals  usually  have  the  form  of  thin 
lamellae  parallel  to  this  plane.  A  mass  consisting  of  many  such 
lamellae  is  said  to  be  "  polysynthetic " ;  an  example  would  be  the 
calcite  described  on  pages  285-286. 

*  Such  a  case  is  explained  in  footnote  f  on  the  opposite  page.  In  this  case 
the  two  equilibria  have  equal  stability. 


2QO  INFLUENCE   OF   OTHER   PROPERTIES 

The  optical  properties  of  such  masses  have  already  been 
treated  in  detail  on  pages  214-219.  It  was  there  shown  that  in 
case  the  twinned  lamellae  are  of  sub-microscopic  dimensions  the 
combination  must  behave  like  a  single  crystal,  whose  properties 
depend  on  the  character  and  orientation  of  the  lamellae,  and 
shown  further  that  when  the  lamellae  are  unequal  in  distribution 
and  thickness  the  behavior  is  optically  anomalous.  (See  p.  219.) 


APPENDIX 
SUPPLY    HOUSES 

FOR 

APPARATUS,  MODELS,  CRYSTALS,  AND 
PREPARATIONS 


291 


[The  price  lists  in  Professor  von  Groth's  book  have  proved  to  be  so  generally 
useful  that  the  translator  deems  it  worth  while  to  include  similar  notices  here, 
no  prices  being  given,  however.  The  number  of  firms  listed  has  been  somewhat 
increased,  but  no  mention  made  of  any  except  German  and  American  houses.] 


292 


APPENDIX 
SUPPLY    HOUSES 

FOR 

APPARATUS,  MODELS,  CRYSTALS,  AND 
PREPARATIONS 


W.  APEL  (Inhaber  DR.  M.  APEL),  Universitats-Mechanikus, 
Gottingen,  Germany. 

Total-reflectometer  after  F.  Kohlrausch  in  three  different  styles,  with  various 
accessory  apparatus.  Also  a  reflection  goniometer  after  Wollaston.  Illus- 
trated catalog. 

BAUSCH  &  LOME  OPTICAL  CO.,  Rochester,  New  York,  U.  S.  A. 

Crystal  refractometer  (Zeiss  manufacture  —  see  p.  297)  specially  adapted  for 
crystallographical  and  mineralogical  investigations. 

Petrographical  microscope  in  a  large  and  a  small  size.  Chemical  microscope  for 
all  branches  of  micro-chemistry.  Projection  apparatus  for  all  classes  of  pro- 
jection, including  micro-projection.  Catalogs. 

BOHM  &  WIEDEMANN,  Chem.-physik.   Utensilienhandl.  u.  mech.  Werkstdtte, 
Karlsplatz  14,  Munich,  Germany. 

Ray-surface  models  corresponding  in  axial  ratios  to  the  index-surface  models 
supplied  by  G.  J.  Pabst  (see  p.  295);  constructed  in  brass  wire  on  a  lacquered 
cast-iron  stand  (or  in  half-size  on  a  wooden  stand)  after  the  specifications  of 
Prof.  v.  Groth  for  lecture  demonstrations:  Ray-surface  of  the  positive  and  of  the 
negative  uniaxial  crystals  and  of  the  biaxial  crystals;  ray-surface  of  the  mono- 
clinic  crystals  for  red,  yellow,  and  blue  represented  by  the  principal-section 
curves,  for  the  demonstration  of  inclined  dispersion;  the  same  to  demonstrate 
horizontal  and  crossed  dispersion;  the  same  to  demonstrate  the  dispersion  in 
a  triclinic  crystal.  Photographs  of  all  the  models  supplied  at  50  pf.  each. 

Apparatus  to  demonstrate  and  determine  the  vibration  directions  of  polarized 
light  in  biaxial  crystals.  (See  footnote  on  p.  154.)  Also  a  reflection  goni- 
ometer. Price  list. 

293 


294  APPENDIX 

THE  C.  H.  COWDREY  MACHINE  WORKS,  Fitchburg,  Massachusetts,  U.  S.  A. 

The  Dwight  Petrotome,  a  machine  for  cutting,  trimming,  and  polishing  rocks, 
minerals,  etc.,  in  rapidly  adjustable  positions  and  for  the  preparation  of  thin 
sections.  Descriptive  pamphlet. 

EIMER  &  AMEND,  205-211  Third  Ave.,  New  York,  N.  Y.,  U.  S.  A. 

Importers.  Standard  foreign  apparatus  for  crystallographical  and  mineralogical 
determinations.  Crystal  collections  and  specimens.  Price  lists. 

FOOTE  MINERAL  COMPANY,  107  North  igth  street,  Philadelphia,  Pa.,  U.  S.  A. 

Specimens  of  minerals  and  rocks.  Decorative  stones.  Complete  catalog  and 
special  lists. 

R.  FUESS,  Mechanisch-optische  WerkstWen.  Abteilung  I.  Diintherstrasse  8, 
Steglitz  bei  Berlin,  Germany. 

Reflection  goniometers  with  vertical  or  with  horizontal  circle,  and  accessories; 
also  theodolite  goniometers.  Total-reflectometers  and  ref ractometers.  Sources 
of  light  and  monochromators. 

Apparatus  for  polarization  and  for  measuring  axial  angles :  Apparatus  to  demon- 
strate the  phenomena  of  double  refraction  of  light  in  calcite,  polarizing  con- 
trivances, polarization  apparatus,  axial  angle  apparatus,  universal  apparatus 
for  crystallographic-optical  investigations,  absorption  of  light  in  crystals, 
preparations  and  utensils  for  interference  phenomena. 

Microscopes  for  physical  and  mineralogical  studies:  Microscopes  with  rotatable 
stage  and  fixed  nicols,  with  fixed  stage  and  rotatable  nicols,  and  for  edu- 
cational purposes;  also  photo-micrographical  apparatus. 

Apparatus  to  demonstrate  the  influence  of  mechanical  forces  on  crystallized  and 
amorphous  bodies.  Cutting  and  grinding  machines,  as  well  as  accessory 
utensils  for  making  rock  and  mineral  preparations.  Projection  apparatus 
with  contrivances  for  projecting  experiments  in  polarized  light.  Catalog. 

ELISHA  T.  JENKS,  Lock  Box  274,  Middleborough  (Plymouth  Co.),  Mass.,  U.  S.  A. 

Special  machines  for  cutting  and  polishing  minerals  and  rocks,  and  for  preparing- 
thin  sections.  Made  only  to  order,  at  various  prices,  and  can  be  changed,  if 
desired,  to  meet  special  wants.  Descriptive  pamphlet. 

GUSTAVUS  D.  JULIEN,  3  Webster  Terrace,  New  Rochelle,  New  York,  U.  S.  A. 

Lapidary  lathes  for  slicing,  grinding,  and  polishing  minerals,  etc.,  and  for  pre- 
paring thin  sections.  Diamond  and  emery  slicing  disks;  grinding  and 
polishing  laps.  Supplies  for  preparing  and  mounting  thin  sections. 

DR.  F.  KRANTZ,  Rheinisches  Miner'alien-Contor  u.  Lehrmittelfabrik,  Herwarth- 
strasse  36,  Bonn  am  Rhein,  Germany. 

Collection  of  four  glass  models  illustrate  the  dispersion  in  orthorhombic  and 
monoclinic  crystals,  with  silk  thread  of  different  colors  drawn  through  to 
represent  the  position  of  the  optic  axes  and  bisectrices.  Rhombohedron 
made  up  of  plate-glass  to  demonstrate  the  double  refraction  in  calcite,  as 
constructed  by  Prof.  K.  Busz.  Glass  model  of  the  Nicol  prism,  as  con- 
structed by  Prof's  Busz  and  K.  Vrba.  Collections  of  models  to  illustrate 
the  optical  relations  of  various  crystals,  as  constructed  by  Prof's  Grubenmann 
and  Weinschenck. 


APPENDIX  295 

Six  gypsum  models  of  the  ray-surface  of  crystals.  Also  six  colored  gypsum 
models  of  the  ray-surface,  as  constructed  by  Prof.  L.  Duparc;  can  be  taken 
apart  along  the  three  principal  sections. 

Three  wooden  models  of  the  optical  index-surface  for  uniaxial  and  biaxial  crys- 
tals according  to  Prof.  P.  von  Groth;  made  of  hard  wood  and  polished;  can 
be  taken  apart.  Also  wooden  models  of  the  index-surface  to  explain  the 
theory  of  double  refraction  and  the  optical  properties  of  crystals;  constructed 
according  to  Prof.  Duparc.  Are  made  of  pear  wood  and  mounted  on  three 
mutually  perpendicular  metal  axes;  are  cut  through  along  properly  oriented 
planes  as  an  aid  to  explaining  the  laws  of  light  transmission. 

Three  wire  models  of  the  optical  index-surface  for  uniaxial  and  biaxial  crystals 
after  Prof.  K.  Vrba;  constructed  in  3  mm.  metal  wire  and  lacquered,  equiva- 
lent directions  with  the  same  color.  Also  four  wire  models  of  the  ray- 
surface  after  Prof.  Vrba,  the  surfaces  for  the  different  rays  being  in  different 
colors. 

Minerals  both  singly  and  in  classified  collections.  Natural  crystals,  oriented 
crystal  sections,  and  thin  sections  of  minerals. 

All  kinds  of  crystallographical  models  prepared  according  to  specification.  Stands 
for  models.  Various  catalogs. 

ERNST  LEITZ,  Optical  Works  in  Wetzlar,  Germany. 
New  York,  30  East  i8th  St.;  Chicago,  1923  Ogden  Ave. 

Eight  different  stands  of  mineralogical  microscopes;  various  optical  outfits  with 
each  stand.  Also  a  mineralogical  demonstration  microscope,  supplied  with 
either  of  two  outfits.  Catalog  in  English.  London  office  9-16  Oxford  street. 

VALENTIN  LINHOF,  Optisch-mechaniscke  Werkstatte,  Goethestrasse  36, 
Munich,  Germany. 

Reflection  goniometer  with  vertical  divided  circle,  or  with  horizontal  divided 
circle  and  two  telescopes. 

G.  J.  PABST,  Lehrmittelfabrik,  Solgerstrasse  16,  Nuremberg,  Germany. 

Index-ellipsoid  models  in  polished  pear  wood,  which  can  be  taken  apart:  The 
index-surface  for  biaxial  crystals,  having  a  circular  and  an  oblique  section, 
together  with  a  wire  model  (see  p.  153,  Fig.  76);  and  the  same  surface  for  + 
and  for  —  uniaxial  crystals,  with  one  oblique  section.  Each  of  the  three 
models  is  made  in  a  large  and  a  small  size.  Catalog. 

A.  H.  PETEREIT,  81-83  Fulton  street,  New  York,  N.  Y.,  U.  S.  A. 
Specimens  of  minerals  and  gems.     Descriptive,  illustrated  price  lists. 

MARTIN  SCHILLING,  Verlagsbuchhandlung,KsLntstrasse  12,  Leipzig,  Germany. 

Gypsum  models:  Ray-surface  of  the  optically  negative  uniaxial  crystals,  with  a 
sector  removed  from  the  ellipsoid,  revealing  the  sphere;  the  same  surface  for 
optically  positive  uniaxial  crystals,  similarly  arranged;  and  the  same  for 
optically  biaxial  crystals,  arranged  so  that  the  outer  surface  can  be  taken 
apart  along  a  principal  section.  Also  the  ellipsoid  corresponding  to  the 
latter  surface,  with  the  same  axes;  as  well  as  the  ray-surface  of  optically 
biaxial  crystals  in  single  octants,  having  on  each  skin  the  curves  of  intersec- 
tion with  spheres  and  with  ellipsoids. 


296  APPENDIX 

Cardboard  models:  Two  triaxial  ellipsoids  made  up  of  22  and  30  circles  respec- 
tively, and  flexible,  so  that  the  axial  ratios  can  be  varied  at  will.  Illustrated 
catalog. 

W.  &  H.  SEIBERT,  Optisches  Institut,  Wetzlar,  Germany. 

Microscopes  for  petrographical  and  crystallographical  investigations.  Six  stands, 
with  different  outfits,  making  eleven  models.  Catalog. 

DR.  STEEG  &  REUTER,  Optisches  Institut,  Homburg  v.  d.  Hohe.  Germany 

Norrenberg  polariscope  having  a  large  field,  and  the  same  with  goniometer  for 
measuring  axial  angles.  Various  forms  of  polarization-microscope,  and  the 
same  with  Lehmann's  heating  apparatus.  Tourmaline  tongs.  Dichroscope, 

A  large  stock  of  oriented  sections  of  natural  and  artificial  crystals.  For  example, 
(i)  uniaxial  crystal  plates  _L  to  the  optic  axis,  including  apophyllite  from 
different  localities,  parti-colored  Brazilian  amethyst,  dextro-  and  levo-quartz, 
calcite  1  to  5  mm.  thick,  apatite,  tourmaline  in  various  colors,  and  magne- 
sium platmo-cyanide;  (2)  biaxial  crystal  plates  JL  to  the  acute  bisectrix, 
including  sanidine  with  different  dispersion,  adularia  (horizontal  dispersion), 
aragonite,  borax  (crossed  dispersion),  brookite  (-L  axial  planes  for  red  and 
blue),  glauberite,  potassium  bichromate  (asymmetric  dispersion),  copper 
sulphate,  gypsum  (inclined  dispersion)  mounted  for  heating,  titanite.  ce- 
russite,  topaz  with  different  axial  angles,  and  cane  sugar;  (3)  pleochroic 
parallelepipeds  mounted  so  as  to  be  rotatable,  including  penninite.  cordierite, 
tourmaline,  topaz,  and  epidote,  as  well  as  andalusite  and  epidote  _L  to  an 
axis  (absorption-brushes);  (4)  collections  of  typical  crystal  plates  in  numbers 
from  fifteen  to  one  hundred. 

Various  other  apparatus  and  preparations,  as  nicols  and  all  kinds  of  polarization- 
prisms,  quartz  prisms  with  the  refracting  edge  II  or  J_  to  the  axis,  calcite 
preparations,  quartz  or  gypsum  wedges,  quarter-undulation  mica  plates, 
gypsum  plates  giving  red  of  the  ist  order,  dextro  and  levo  Reusch  mica 
combinations,  cooled  glasses,  crystal  plates  with  absorption-stripes,  pocket 
spectroscope,  apparatus  for  observing  conical  refraction,  apparatus  and 
preparations  for  investigating  piezo-electricity,  etc.,  cutting  and  grinding 
machines  in  various  sizes.  Detailed  catalog. 

PETER  STOE,  Mechanische  Werkstatte,  Jubilaums-Platz  70,  Heidelberg,  Germany. 

Two-circle  reflection  goniometer  as  constructed  by  Goldschmidt,  in  two  different 

models,  with  lamp. 
Grinding  apparatus  after  Wulfing.     Price  list,  with  illustrations. 

W.  HAROLD  TOMLINSON,  Petrological  Laboratory,  44  E.  Walnut  Lane, 
Germantown,  Philadelphia,  Penna.,  U.  S.  A. 

Microscopic  sections  of  minerals,  rocks,  etc.,  for  colleges  and  schools.  Oriented 
crystals.  Thin  sections  prepared.  Samples  examined  microscopically  and 
reported  upon.  Catalog. 

VOIGT  &  HOCHGESANG  (Inhaber  M.  R.  BRUNEE),  Fabrik  fur  Diinnschli/c 
von  Gesteinen,  Mineralien,  etc.    Untere  Maschstrasse  26,  Gottingen,  Germany. 

Collections  of  thin  sections  of  minerals:  Large  educational  collection  consisting 
of  115  thin  sections  of  the  petrographically  most  important  minerals,  the 
crystallographic  orientation  being  arranged  with  special  reference  to  deter- 


APPENDIX  297 

mination  of  the  crystal  system;  the  material  was  chosen  and  classified  by 
Prof.  C.  Klein,  who  also  examined  the  collection  employed  as  a  model  in 
preparing  the  others.  Smaller  educational  collection,  for  beginners  and 
schools:  fifteen  thin  sections  for  the  microscopical  demonstration  of  impor- 
tant properties  of  crystals;  arranged  by  Prof.  F.  Rinne.  Special  collection  of 
twenty-five  thin  sections  for  demonstrating  optical  anomaly  in  some  of  the 
crystals  of  the  isometric  system;  modeled  after  a  collection  arranged  by 
Prof.  C.  Klein. 

Crystal  plates  of  all  kinds;  special  collection  of  nine  polished  crystal  plates  for 
demonstrating  circular  polarization,  as  well  as  dispersion  of  the  optic  axes, 
the  bisectrices,  and  the  optic  axial  plane.  Also  various  collections  and 
specimens  of  thin  rock  sections.  Supplies  and  utensils  for  the  preparation 
of  thin  sections.  Oriented  sections  prepared  from  crystals  in  directions  as 
prescribed;  thin  sections  from  rocks  as  desired.  Catalog;  small  catalog 
in  English. 

WARD'S  NATURAL  SCIENCE  ESTABLISHMENT,  76-104  College  Ave., 
Rochester,  New  York,  U.  S.  A. 

Collection  of  crystals  to  illustrate  the  optical  properties  prepared  from  stock. 
Foreign  collections  imported.  Catalogs  and  price  lists. 

CHARLES  L.  WHITTLE,  50  Congress  street,  Boston,  Mass.,  U.  S.  A. 

Diamond  saws  for  slicing  minerals  and  rocks.  May  be  used  with  special  machin- 
ery or  with  any  ordinary  lathe.  Made  in  regular  sizes  and  in  special  sizes  as 
desired.  Descriptive  pamphlet. 

CARL  ZEISS,  Optische  Werkstatte,  Jena,  Germany. 

Crystal  refractometer  (new  Abbe  model  after  Pulfrich),  specially  adapted  for 
crystallographical  and  mineralogical  investigations.  Microscopes  in  two 
large  models  for  mineralogical  and  crystallographical  investigations,  and  in 
two  small  models  for  educational  purposes.  Catalogs  in  English.  London 
office  W,  29  Margaret  street,  Regent  street. 


Fig.  /  Calcite 


.  S.  Brook  its 


Fiq's  3  4. 4*4ragonite 


fig's  6&  7.  Gypsum 


Verlag  von  Wilhelm  Engelmann  in  Lei 


Fig 's  £&2  Samc/ine 


Fig  ;y/#  &  77.j5or.yx 


M.  F.  Jutte,  Kunstanstalt,  Leipzig. 


INDEX 


No  references  to  the  Appendix  are  included. 


Abbe's  refractometer,  42,  150 
Abnormal  interference-colors,  69,  109 

dispersion  of  the  optic  axes,  195 
Absorbency,  237 
Absorption  axes,  241 
coefficient  of,  234 
colors  of  biaxial  crystals,  241-245 

of  uniaxial  crystals,  236-241 
formula,  234 

influence  of  strain  on,  270 
of  light,  25,  232-252 

in  biaxial  crystals,  241-245 
in  singly  refracting  crystals,  235 
in  uniaxial  crystals,  235-241 
spectrum,  233 
surface,  Mallard's,  235 
tufts,  246 

Acentric  properties,  6 
Achromatic  retardation  plates,  199 
Activity  [see  Optical  activity] 
Adams's  polarization  and  axial  angle 

apparatus,  194 
Aggregate,  crystalline,  71 
polarization,  71 
radiating-fibrous,  optical  behavior  of, 

72-74,  247 
Airy's  spirals,  230 
Allochromatic  coloring,  233 
Alum,  281 

Ammonium  mellitate,  172 
Amorphous  bodies,  3,  4,  21 
polarization  in,  74 
variation  of  optical  properties  by 

strain,  263-270 
Amplitude  of  vibration,  12 
Analyzer,  58 
circular,  231 


Andalusite,  Brazilian,  246 
Angle,  axial  [see  Axial  angle] 
critical,  36 
measurement,  28-30 
of  extinction,  76 
of  incidence,  27 
of  reflection,  28 
of  refraction,  32 
polarizing,  49 
Anglesite,  257 
Anisotropic  body,  21,  51 
Anomalies,    optical,   72,   219,   279-282, 

290;  denned,  280 

Mallard's  explanation  of,  219,  281 
true,  282 
Apatite,  252 
Apophyllite,  118 
Apparent  axial  angle,  186 
Arago,  F.,  223 
Aragonite,  257 
Arzruni,  A.,  257 
Asymmetric  dispersion,  182 
Attractive  crystals,  103 
Axes,  absorption,  241 
electric,  283 
of    optical   elasticity   [see  Principal 

vibration  directions] 
optic  [see  Optic  axes] 
principal,  of  an  ellipsoid,  126 
of  strain,  263 
thermic,  259 

Axial  angle  [see  Optic  axial  angle] 
apparatus,  Adams's,  194 
apparent,  186 
colors,  242 

figure,  110-120,  161-183 
of  rotatory  crystals,  229-230 


299 


300 


INDEX 


Axis  of  incidence,  27 
of  rhombohedron,  90 
optic,  92  [see  also  Optic  axis] 

Babinet's  rule,  236 
Barite,  257 
Baumhauer,  H.,  287 
Beaulard,  F.,  277 
Becke's  method,  45 
Ben  Saude,  A.,  281 
Beryl,  255 

Biaxial  crystals,  121-195;  defined,  129 
absorption  in,  241-245 
determination  of  optical  character 

of  [see  Determination] 
index-surface  of,  126-129 
influence  of  heat  in,  256-260 

of  strain  in,  278-279 
interference     phenomena    of    [see 

Interference  phenomena} 
negative,  138,  144 
optical  activity  in,  231 

constants  of,  151 
positive,  137,  143 

principal  refractive  indices  of,  126 
[see  also  Determination,  Varia- 
tion] 

vibration  directions  of,  126  [see 
also  Determination,  Disper- 
sion, Variation] 
ray-surface  of,  129-144 
Binary  axis,  181 
Bi-normal  cone,  155 
Bi-normals,  142 
Biot,  223,  225 
Biot's  quartz  plate,  228 
Bi-radials,  132 
Birefringence,  63,  101  [see  also  Double 

refraction] 

determination  of,  63 
of  calcite,  101 

unequal  for  different  colors,  69,  109 
variation  of  [see  Variation] 
Birefringent  crystals,  54 

behavior  of,  in  polarized  light,  59-62 


Birefringent  substances,  53 
Bisectrices,  dispersion  of,  174-183  [see 

also  Dispersion] 
Bisectrix,  acute,  138 

negative,  211 

obtuse,  138 

positive,  2ii 
Bi-vector  properties,  4 

of  higher  symmetry,  6,  7 
of  lower  symmetry,  6,  7 
Body  color,  234 
Bolzmann's  formula,  221 
Borax,  181 

Brewster,  D.,  234,  270,  281 
Brightness  of  ray,  14 
Brookite,  170,  172-173 
Brush  phenomena,  246-248 
Bucking's  compression  apparatus,  274 
Biitschli,  74 

Calcite,  55,  81-101,  238,  252,  255,  285, 

287,  289 

rhombohedron,  81 
Cauchy,  A.,  250 
Cauchy's  dispersion  formula,  46 

constants  of,  106,  151 
Celestite,  257 

Center  of  symmetry,  6,  180 
Centrically  symmetrical  properties,  6 
Character    of     double    refraction    [see 
Determination  of  optical  char- 
acter] 

opposite,  109 

Characters  of  light  rays,  87,  89,  122 
Chaulnes's  method,  44 
Chemical  crystallography,  5 
Chlorite,  240 
Circular  analyzer,  231 

double  refraction,  laws  of,  223-224 
polarization,  20,  220,  223 
by  total  reflection,  50 
section,  123,  128 
vibrations,  16,  220 

Classification  of  crystal  properties,  3-7, 
253 


INDEX 


301 


Classification  of  crystals  with  reference 

to  elastic  strain,  272-273 
with  reference  to  optical  properties, 

196-197 
Coefficient  of  absorption,  234 

of  extension,  271 
Cohesion  of  crystals,  7,  262 
Colloid  substances,  268 
Color,  body,  234 
complementary,  62 
in  reflected  light,  234 
in  transmitted  light,  234 
of  light  ray,  14 
scale,  Radde's,  245 
schiller,  249 
surface,  249 

Coloring,  allochromatic,  233 
dilute,  233 
idiochromatic,  233 
Colored  and  colorless  substances,  232 

glasses  for  standard  colors,  245 
Colors  of  crystals,  233-246,  249-251 
axial,  242 

interference    [see    Interference-colors] 
of  the   ist,  2d,  3d,   and  4th  orders, 

63-69  [see  also  Interference-colors] 
polarization  [see  Interference-colors] 
standard,  245 

Combination    (interference)    of    plane- 
polarized  light,  16-20,  54,  58 
Combinations,  mica   [see  Mica  combi- 
nation] 
of  doubly  refracting  crystals,  optical 

behavior  of,  198-220 
Common  light,    16   [see  also  Ordinary 

light] 

Compact  minerals,  71 
Complementary  color,  62 
Compression,  influence  of,  in  crystals, 
231,  261-262,  273-279  [see  also 
Variation] 
in  glass,  263-268 
Condensing  lens,  no 
Conical  refraction,  exterior,  141 
interior,  142 


Conoscope,  77-80 

Constants  of  Cauchy's  formula,   106, 

151 

of  elasticity,  271 

optical,  of  biaxial  crystals,  151 
Convergent  light,  investigations  in,  77 
Cooled  glasses,  268,  270 
Copper  sulphate,  182 
Cordierite,  244,  257 
Critical  angle  of  total  reflection,  36 
Crossed  dispersion,  181 

axial  planes  for  different  colors,  172 

resulting  from  pressure,  279 
Crystal   form,   relation   of,    to   optical 
properties,  4,  81,  183,  197 

structure,  5 
Crystalline  aggregate  [see  Aggregate] 

body,  3 

Crystallization,  3 
Crystallized  body,  3 
Crystallography,  chemical,  5 

physical,  4 

Cubic  crystal  system,  197,  273 
Curie,  J.  and  P.,  283 

Darkness,  position  of,  61 

Deformation,  261  [see  also  Strain] 

Des  Cloiseaux,  258 

Determination  of  birefringence,  63 
of  faster  and  slower  ray,  198-204 
of  optic  axes  in  biaxial  crystals,  183- 

iQS 

of  optic  axial  angle,  143,  183-195 
of  optic  axis  in  uniaxial  crystals,  106 
of  optical  character,  198-210 

of  biaxial  crystals,  198-204,  209- 

210 

of  uniaxial  crystals,  198-209 
with  gypsum  plate,  200-201 
with  mica  plate,  199-200,  205- 

210 

with  quartz  wedge,  201-204 
of  optical   constants   by   Miittrich's 

method,  194 
of  order  of  interference-color,  201 


302 


INDEX 


Determination  of  principal  refractive 
indices  of  biaxial  crystals,  144- 
152 

of  uniaxial  crystals,  104-106 
of  principal   vibration  directions  of 

biaxial  crystals,  151 
of  refractive  index,  38-45,  118 
of  vibration  directions  of  a  crystal 

plate,  76,  154,  158 
Dextro-crystal,  225 
Diamond,  48,  254,  280 
Dichroic  fluorescence,  252 
Dichroism,  237,  244 
by  elastic  strain,  270 
temporary,  270 
Dichroite,  245 
Dichroscope,  238 
Dichroscopic  lens,  238 
Dielectric   crystals,   influence   of   elec- 
trization in,  283 
Difference  in  phase,  17 

of  path,  17 
Dilute  coloring,  233 
Dimorphism,  5 
Dispersion  formula,  Cauchy's,  46 

constants  of,  106,  151 
for  dispersion  of  the  axes,  195 
models,  174 

of  bisectrices,  174-183  fsee  also  Dis- 
persion   of    principal    vibration 
directions] 
of  light,  45 

variation  of  [see  Variation} 
of  optic  axes,  170-171 
abnormal,  195 
formulae  for,  195 
of  principal  vibration  directions,  158- 

160,  173-183 
asymmetric,  182 
crossed,  181 
horizontal,  179 
incb'ned,  177 
Dispersive  power,  46 
Disturbance    in    crystal     growth,    in- 
fluence of,  72 


Double  refraction  of  light,  50-80;  de- 
fined, 53 

by  metallic  films,  251 
character  of  [see  Optical  character] 
in  calcite,  81-101 
in  other  uniaxial  crystals,  101-106 
of  opposite  character,  109 
strength  of,  63,  101  [see  also  Bire- 
fringence] 
through  compression  and  extension, 

263-270 
variation     of     [see     Variation    of 

birefringence] 

weak:  how  to  recognize  it,  199-200 
Doubly  refracted  rays,  53 
Doubly  refracting  crystals,  54 

behavior  of,  in  polarized  light, 

59-62 

substances,  53 
Dove's  experiment,  90 
Drude,  P.,  250 
Dufet,  231,  255,  257 

Elastic  limit,  261 

strain,  261-283;  defined,  261 
by  electrical  action,  282-283 
by  mechanical  forces,  261-282 
homogeneous,  261-262 
influence    of,     262-283    [see    also 

Variation] 

not  homogeneous,  262-279 
Elasticity,  7,  261,  262  ;  defined,  261 
axes  of   optical  [see  Principal  vibra- 
tion directions] 

classification  of  crystals  with  refer- 
ence to,  272-273 
constants  of,  271 

ellipsoid  of  optical  [see  Index-surface] 
of  the  ether,  n 
Electric  axes,  283 
Electrical  action,  elastic  strain  by,  282- 

283 

influence  of,  283 
properties,  7 
Electro-magnetic  theory  of  light,  89,  254 


INDEX 


303 


Ellipsoid,  index  [see  Index-surface] 

inverse,  129 

of  elasticity,  129 

of  equal  work,  129 

of  indices,  1 29 

of  polarization,  129 

of  rotation,  93 

of  strain,  263 

triaxial,  6,  126 
Ellipsoidal  properties,  7 
Elliptical  polarization,  20,  215 
by  total  reflection,  50 

vibrations,  16 
Emerald,  255 
Epidote,  245,  246 
Epoptic  figures,  246 
Ether  [see  Luminiferous  ether] 
Expansion  by  heat,  influence  of,  253,  254 
Extension,  coefficient  of,  271 

influence  of,  in  glass,  264-267 
Extension  coefficients,  surface  of,  271 
Exterior  conical  refraction,  141 
Extinction  angle,  76 

position  of,  61 
Extraordinary  ray,  82 

Facial  color,  243 

Faster  and  slower  ray,  determination 

of,  198-204 
Feldspar,  179 
Field,  79 
Fizeau,  254,  255 
Fletcher,  L.,  n 
Flowing  crystals,  284 
Fluorescence,  251 

dichroic,  252 

Fluorite  (fluor-spar),  251,  254 
Fraunhofer's  lines,  47-48 
Fresnel,  A.,  135,  223 
Fresnel's  rhomb,  49-50 

surface,  135,  150 
Front-normal,  25 

Galena,  250 
Gehlenite,  no 


Gelatine,  optical  behavior  of,  263,  268- 
270 

models  of  crystals:    optical  behavior 

of  sections,  281 

Geometry,   relation    of,  to    crystallog- 
raphy, 8 

Glass,  influence  of  strain  in,  263-268 
Glasses,  cooled,  268,  270 

colored,  for  standard  colors,  245 
Glauberite,  259 
Glaucophane,  245 
Gliding  and  its  influence,  284-286 

lamellae,  286 

planes,  284 
Gold,  250 

Goniometer,  reflection,  28-30 
Gouy,  M.,  223,  277 
Grazing  incidence,  method  of,  44 
Growth  of  crystals,  7,  72,  288 
Guye,  C.  E.,  221 

Gypsum,  176,  177,  195,  211,  257,  259, 
260 

lamellae  in  layers,  211-220 

plate,  66,  198,  200,  210 
use  of,  200-201 

Haidinger,  W.,  234,  249 
Haidinger's  brushes,  247 

lens,  238 

Halite  (rock-salt),  280,  284,  285 
Heat,  influence  of,  253-260 
Heating  apparatus,  258 
Helmholtz,  H.  L.  F.  v.,  248 
Herapathite,  54 
Heterogeneous  media,  25 
Heterotropic  media,  21 
Hexagonal  crystal  system,  197,  273 
Homeotropy,  284 
Homogeneous  body,  3 

light,    47    [see    also    Monochromatic 
light] 

strain,  261-262 
Horizontal  dispersion,  179 
Huygens.  C.,  n,  93,  101 
Huygens's  construction,  24 


3°4 


INDEX 


Huygens's  construction,  application  of, 

26,  32,  92,  95,  103 
Hydrogen,  line  spectrum  of,  47,  48 

Iceland  spar,  55 
Idiochromatic  coloring,  233 
Incidence,  angle,  axis,  plane  of,  27 

grazing,  method  of,  44 
Inclined  dispersion,  177 
Index  of  refraction,  33  [see  also  Refrac- 
tive index] 
surface    (indicatrix,    etc.),    121-129; 

denned,  123 
models,  135 
of  biaxial  crystals,  126 
of  singly  refracting  crystals,  125 
of  uniaxial  crystals,  94,  123 
variation  of  [see  Variation] 
Indicatrix,  n,  123,  129  [see  also  Index- 
surface] 

Interference  colors,  63-71 
variation  in,  69,  109 
figure,  79  [see  also  Interference  phe- 
nomena] 
of  plane-polarized  light,    16-20,   54, 

58;  defined,  16 
phenomena  due  to  absorption,  246- 

248 
of  biaxial   crystals  in   convergent 

light,  161-183 
in  parallel  light,  152-161 
of    combinations    [see     Combina- 
tions] 

of  rotatory  crystals,  220-231 
of  uniaxial  crystals  in  convergent 

light,  1 10-120 
in  parallel  light,  106-110 
Interior  conical  refraction,  142 
Inverse  ellipsoid,  129 
lodo-quinine  sulphate,  54 
Iron,  250 

Isochromatic  curves,  116,  161 
Isomerism,  chemical,  5 

physical,  5 
Isometric  crystal  system,  197,  273 


Isomorphic,-ous  substances,  6 
mixtures,  6 

optical  behavior  of,  282 
Isotropic  bodies,  3 

optically    [see    Optically    isotropic 
bodies] 

Kalkowsky,  E.,  156 

Klein's  rotation  apparatus,  185 

Klocke  and  Brauiis,  282 

Kohlrausch,  W.,  144,  150 

Kohlrausch,  F.:  total  reflectometer,  40 

[see  also  Totd-reflectometer] 
Kundt,  251,  270,  283 

Lang,  V.  v.,  238 
Lead,  250 
Lehmann,  O.,  284 
Lemniscates,  163,  166 
Lenses,  effect  of  pressure  in,  267 
Leucocyclite,  119 
Levo-crystal,  225 
Light,  dispersion  of,  45 

hydrogen,  47,  48 

lithium,  47 

monochromatic,  47-48 

nature  of,  11-16 

ordinary,  16,  90 

polarization  of  [see  Polarization] 

polarized  [see  Polarized  light] 

propagation  of,  21-26 

ray,  14,  24 

reflection  of,  26-30 

refraction  of,  30-48 

sodium,  47,  48 

sun's,  47-48 

thallium,  47 

undulatory  theory  of,  11-14 

velocity,  14,  15 

vibrations,  12,  15,  16 

white,  45 

Limit  of  solidity,  262 
Line  spectrum  of  hydrogen,  47,  48 
Liquid  crystals,  284 
Lithium  light,  47 


INDEX 


305 


Lommel,  251 
Luminiferous  ether,  n,  21 

changed  through  heat,  253 
through  strain,  262 

elasticity  of,  n 

resilience  of,  n 

Mach  and  Merten,  277 

Magnesium  platino-cyanide,'   241,  249, 

251 

Magnetic  properties,  7 
Mallard's  absorption-surface,  235 

explanation  of  anomalies,  219,  281 
Mean  line,  138  [see  also  Bisectrix] 
Measurement  [see  Determination] 
Mechanical  forces,  action  of,  8 

influence  of,  261-282,  283-288 
Metallic  films,  double  refraction  in,  251 
reflection,  249,  250 

influence  of  heat  on,  254 
Mica,  198-199,  211 

combination,  Norrenberg's,  213 

Reusch's,  217,  218 
lamellae  in  layers,  211-220 
plate  (i  undulation),  198,  210 

use  of,  199-200,  205-210 
Microscope,  polarization,  75,  80 
use  as  conoscope,  80 
as  orthoscope,  75 
for  determining  refractive  indices, 

44,  45 
Minimum  deviation  of  light,  37 

method  of,  39 
Mirror-plane,  28 
Monochromatic  light,  47 

how  obtained,  47-48 
Monoclinic  crystal  system,  197,  273 
Morphotropic  relationships,  6 
Miittrich's  method,  194 

Negative  biaxial  crystals,  138,  144 

bisectrix,  211 

uniaxial  crystals,  103 
Neumann's  theory  of  elasticity,  263 

experiment,  264-267 


Nicol  prism  (nicol),  55,  98-99 
Norrenberg's        mica        combination, 

^13 
polariscope,  80 

Obtuse  bisectrix,  138 

Offret,  255,  257 

Oligoclase,  257 

Opaque  bodies,  25,  232 

Optic  axes  of  biaxial  crystals,  129,  132, 

142 

determination  of,  183-195 
primary,  129,  132,  142 
secondary,  132 
axial  angle,  143 

calculation,  143 
measurement,  183-195 
resulting  from  strain,  269,  270, 

274,  276,  280 
variation  [see  Variation] 
axis  of  uniaxial  crystals,  92 

determination  of,  106 

rotated  through  pressure,  288 

Optical  activity,  220  [see  also  Rotation 

of  polarization  plane] 
anomalies  [see  Anomalies] 
character  (sign)  [see  Determination] 
constants,  151 

determination  of    [see  Determina- 
.     lion] 

of  twinned  lamellae,  220 
properties,  7,  14,  84 
rotatory  power  [see  Rotatory  power] 
sign  [see  Optical  character] 
symmetry,  planes  of,  173,  174 
Optically  active  crystals,  220-231 

lists  of,  222-223,  231 
anomalous  crystals  [see  Anomalies] 
biaxial  crystals,  129  [see  also  Biaxial 

crystals] 

isotropic  bodies,  21-50;  defined,  21 
uniaxial  crystals,  101  [see  also  Uni- 
axial crystals] 
Ordinary  light,  16,  90 
ray,  82 


306 


INDEX 


Organic  substances,  optical  behavior  of, 

270 

Orthorhombic  crystal  system,  197,  273 
Orthoscope,  75 
Other  properties,  influence  of,  253-290 

Packets  of  crystal  lamellae,  211-220 
Parallel  light,  investigations  in,  77 
Path  difference,  17 
Penninite,  101,  240 
Percussion  figure,  284 

test,  284 

Period  of  vibration,  12 
Permanent    strain,   influence    of,    280, 

283-288 

Phase,  difference  in,  17 
Phenacite,  255 
Phosphorescence,  252 
Physical  crystallography,  4 

isomerism,  5 
Piezo-electricity,  283 
Plane  of  incidence,  27 

of  polarization,  15 

of  symmetry,  28 

parallel  plate,  37 

polarized  light,  15,  85 
Planes  of  optical  symmetry,  173,  174 
Plasticity  and  its  influence,  280,  283-284 
Platinum,  250 
Pleochroic  halos,  280 
Pleochroism,  234 
Pockels,  F.,  270,  273,  283 
Pocklington,  231 
Polariscope,  59,  74-80 
Polarization  apparatus,  74-80 

brushes,  246 
Haidinger's,  247 

colors,  63-71  [see  also  Variations] 

instrument,  59,  74-80 

microscope,  75,  80 

of  light,  15,  16,  20,  49,  53,  85,  220,  223 
by  reflection  and  refraction,  49-50 

plane,  15 

Polarized  light,  circularly,  16,  20 
elliptically,  16,  20 


Polarized  light,  plane,  15,  53,  85 
Polarizer,  55,  58 
Polarizing  angle,  49 

prisms,  55 
Pole-angle,  287 
Polymorphism,  5 
Polysymmetry,  220 
Polysynthetic  twinning,  220,  289 
Position    of  darkness   (of    extinction), 

61 

Positive  biaxial  crystals,  137,  143 
bisectrix,  211 
uniaxial  crystals,  103 
Potassium  chloride,  254 
Pressure  [see  Compression,  Variation] 
Primary  optic  axes.  129,  132,  142 
Principal  axes  of  an  ellipsoid,  126 

of  strain,  263 
optic  section,  90,  127 
refractive  indices  of  biaxial  crystal? 
126  [see  also  Determination, 
Variation] 
of  calcite,  100 

of  other  uniaxial   crystals,    101 
[see  also  Determination,  Va- 
riation] 
section  of  an  ellipsoid,  126 

of  a  calcite  rhombohedron,  81 
thermic  axes,  259 

vibration  directions  of  biaxial  crys- 
tals, 126  [see  also  Determina- 
tion, Dispersion,  Variation] 
Prism,  37 
Prismatic  dispersion  of  light,  45 

spectrum,  45 
Propagation  of  light,  21-26 

rectilinear,  24 
Pyroelectricity,  283 

Quarter-undulation  plate,  198,  210 

achromatic,  199 

use  of,  199-200,  205-210 
Quartz,   103,   221,   223,   224,   231,  255, 

256,  277 
plate,  Biot's,  228 


INDEX 


307 


Quartz  wedge,  68,  201-202 
use  of,  201-204 

Radde's     color    scale,    application    of, 

245 
Radiating-fibrous     aggregates,     72-74, 

247 

Ray  axes,  132,  142 
characters,  87,  89,  122 
front,  25 
of  light,  14,  24 
surface  (wave-surface),  24 
models,  135 

of  biaxial  crystals,  129-144 
of  calcite,  94 

of  other  uniaxial  crystals,  102-103 
of  singly  refracting  crystals,  22,  24, 

92-93 

of  tartaric  acid,  143,  144,  150 
Rectilinear  propagation  of  light,  24 

vibration,  16,  85 
Reflected  ray,  26 
Reflection  goniometer,  28-30 
metallic,  249,  250 
of  light,  26-30 

law  of,  28 
Refracted  ray,  26 
Refracting  angle,  37 
Refraction,  angle  of,  32 
exponent,  33 
index  of,  33 
influence  of  heat  on,  254-260 

of  strain  on,  273,  279 
of  light,  30-48 

law  of,  33 
quotient,  33 

Refractive  index  (refringency,  etc.),  33 
determination  of,  38-45,  118 

in  microscopic  crystals,  44-45 
of  metals,  250 

relation  of  to  light  velocity,  33 
variation  of  [see  Variation] 
indices,  principal    [see  Principal  re- 
fractive indices] 
Refractometer,  42,  150 


Refringency,    40    [see    also    Refractive 

index] 

Repulsive  crystals,  103 
Resilience,  n 
Resin,  74 

Reusch,  E.  v.,  281,  284,  285,  288 
Reusch's  mica  combination,  217,  218 
Rhombohedron,  calcite,  81 
Rock  section,  75,  108,  120,  160,  245 

salt  (halite),  280,  284,  285 
Rotation  apparatus,  Klein's,  185 
ellipsoid,  93 
of  the  polarization  plane,  218, 220-231 

[see  also  Rotatory  power] 
Rotatory  crystals,  218 

lists  of,  222-223,  231 
power,  218 
opposite,  222 
specific,  221 

variation  of  [see  Variation] 
tables  of,  221,  223 
Rudberg,  257 

Sanidine,  179,  257,  259 
Scalar  properties,  4 
Schiller  color,  249 

oriented,  233 

Secondary  optic  axes,  132 
Senarmont's  salt,  246 
Sense  of  the  dispersion,  171 
Sensitive  tint  (teinte  sensible),  228  [see 
also  Sensitive  violet] 

violet  (sensitive  tint),  66 
Sign  [see  Character] 
Silver,  250 
Singly  refracting  bodies,  21-50;  defined, 

S3 

crystals,  54 
absorption  in,  235 
behavior  of,  in  polarized  light,  59 
index-surface  of,  125 
influence  of  heat  in,  254-255 

of  strain  in,  273 

optical  activity  in,  220-222,  231 
ray-surface  of,  22,  24,  92 


308 


INDEX 


Singly    refracting    crystals,    refractive 
index  of,   33  [see  also  Determina- 
tion, Variation] 
Sodium  chlorate,  221,  255 

chlorid,  254 

light,  47,  48 
Sohncke,  L.,  252 
Solid  solution,  233 
Solution  of  crystals,  7 
Soret  and  Sarasin,  221 
Specific  rotatory  power,  221 
Spectral  decomposition  of  light,  45,  47 
Spectrum,  absorption,  233 

line,  47 

prismatic,  45 

solar,  47-48 
Spheroid,  93 

Spherulite,  optical  behavior  of,  72 
Standard  colors,  245 
Stauroscope,  158 
Stibnite,  250 
Strain  (deformation),  axes  of,  263 

elastic,  261  [see  also  Elastic  strain] 

ellipsoid  of,  263 

homogeneous,  261-262 

permanent  [see  Permanent  strain] 
Strength  of  double  refraction,  63,  101 

[see  also  Birefringence] 
Strontium  nitrate,  246 
Structure,  crystal,  5 

radiating-fibrous,  72,  247 
Sunlight,  47-48 
Surface  colors,  249-251 

of  extension  coefficients,  271 

of  reference,  121-129 
Symmetry,  center  of,  6,  180 

optical,  planes  of,  173,  174 

plane  of,  28 

relation  of,  to  optical  properties,  4, 183 
[see  also  Crystal  form] 

Tartaric  acid,  144,  150 
Temporary  dichroism,  270 
Tension  [see  Extension] 
Tensor  properties,  4 


Tetragonal  crystal  system,  197,  273 
Thallium  light,  47 
Thermal  properties,  7 

influence  of,  253-260  [see  also  Vari- 
ation] 

Thermic  axes,  principal,  259 
Thin  section,  75,  108,  120,  160,  245 
Topaz,  257 
Total  reflection,  36 

critical  angle  of,  36 

reflectometer,  40 

application  of,  40,  103,  148,  193 
Tourmaline,  54,  240 

tongs,  75 

Transparent  bodies,  25,  232 
Transverse  plane,  15 
Triaxial  ellipsoid,  6,  126,  262 
Trichroism,  244 

Triclinic  crystal  system,  197,  273 
Trigonal  crystal  system,  197,  273 
Trilling,  289 
Trimorphism,  5 
Tschermak's  method,  246 
Twin  crystal,  219,  289 

plates,  212 

Twinned  lamellae,  220,  289 
Twinning,  211,  288-289 

arising  by  pressure,  285-286 

influence  of,  214-219,  286,  290 

lamellae,  286,  289 
microscopic,  290 

plane,  289 

striation,  286 

Umbilical  point  of  ray-surface,  141 
Undulatory  theory  of  light,  11-14 
Uniaxial  crystals,  81-120;  defined,  101 
absorption  in,  235-241 
behavior  of,  in  polarization  appa- 
ratus    [see   Interference    phe- 
nomena] 
determination  of  optical  character 

of  [see  Determination] 
index-surface  of,  94,  123 
influence  of  heat  in,  255-256 


INDEX 


3°9 


Uniaxial  crystals,  influence  of  strain  in, 

273-277 
negative,  103 

optical  activity  in,  220-231  [see  also 
Rotatory  crystals,  Rotatory  power] 
positive,  103 

principal  refractive  indices  of,  100- 
101  [see  also  Determination, 
Variation] 

ray-surface  of,  102-103 
Uranium  glass,  251 

Variation  of  axial  angle  by  heat,  258- 

260 

by  strain,  278-279 
of  birefringence  (see  also  Variation  of 

indices}  by  Heat,  257-258 
by  strain,  273 
of  dispersion  by  heat,  254-257 

by  strain,  270 
of  index-surface  by  heat,  260 

by  strain,  270,  279 
of  indices  by  heat,  253-260 

by  strain,  273-279 
of  metallic  reflection  by  heat,  254 
of  principal  vibration  directions  [see 

Variation  of  index-surface] 
of  rotatory  power  by  heat,  254-255, 

256 

by  strain,  231,  2^7 

Variations  in  polarization-colors,  69,  109 
Vector  properties,  4 
Velocity  of  light,  14,  15 
Vibration,  12 


Vibration  circular,  16,  20 

direction  of  polarized  light,  53,  90 

determination  of,  76,  158 
directions  of  a  crystal,  determination 

of,  76,  158 
of  a  biaxial  plate,  153 

determination  of,  76,  154,  158 
dispersion    of,    158    [see    also 

Dispersion] 
elliptical,  16,  20 
period,  12 
plane,  15 
rectilinear,  i6,  85 
Voigt,  W.,  247,  250 

Wave,  15 
length,  13 

in  different  bodies,  15 
of  different  colors,  14,  15 
motion,  13 
plane,  24 

surface,  22  [see  also  Ray-surface] 
system,  14 
Weak  double  refraction,  recognition  of, 

199-200 
Wedge,  quartz,  68,  201-202 

use  of,  201-204 
White  light,  45 

of  a  higher  order,  70 
Wiener,  224 
Wollaston's  method,  42 

Yellow  spot  of  the  retina,  248 
Yttrium  platino-cyamde,  246 


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Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

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Baker's  Roads  and  Pavements 8vo,  5  00 

Treatise  on  Masonry  Construction 8vo,  5  00 

Black's  United  States  Public  Works Oblong  4to,  5  00 

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14 


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14 


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